rules of algebra

8/13/2019 Rules of Algebra

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The Order of Operations

In algebra, there exists the following fundamental operators: Parentheses, Exponents, Division and

Multiplication, Addition and Subtraction. Many algebraic expressions or equations you come across use more

than one operator and their solutions will be different depending on the order you carry out the diffrent operations.

Keep in mind that all expressions or equations are evaluated from left to right.

For example, given the expression:

One possible solution could be 36 - 4 x 6 = 192, if we choose to evaluate the expression as follows:

then

Alternatively, one can also evaluate the expression as follows:

Out of two possible solutions, only the second is correct.

In algebra, any given expression or equation with more than one operator is evaluated using a standard notation

referred to as the order of operations which is represented by the acronym PEMDAS.

PEMDAS

One easy way to remember the order of operations is to use the acronym PEMDAS Observe that PEMDAS isformed using the first letter of each of the above operators. (Parentheses, Exponents, Multiplication

and Division,Addition and Subtraction).

Order of Operations is a standard used to denote which operators have precedence over the others and thus are

solved first in any equation or expression.

Examples of the Order of Operations


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Example 1

Evaluate the following

Step 1

First, evaluate whatever is in the parentheses:

Step 2

Next, evaluate the exponent:

Step 3

Evaluate any multiplication and division from left to right:

Step 4

Evaluate any addition and subtraction from left to right or which ever way makes it easier for you:

note that

is evaluated as

Example 2

Solve forx in the equation below

Step 1


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As always evaluate the expression within the parentheses first, since there is more than one operator in the

parentheses, apply PEMDAS to the expression

Step 2

Divide both sides by 21

Step 3

Example 3

Evaluate the following

Step 1

note that in the above expression, one can choose to divide first since that makes the computation easier

Step 2

Example 4

Solve forx in the equation below


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Step 1

Step 2

Step 3

Step 3

Distributive Property


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The Distributive Property is a useful algebra property that simplifies multiplication. Before we define the Distributive

property, let us first revisit Commutativity.

Commutativity is an algebra property that says that changing the order with which a computation is done does not

change the end result. Commutativity applies mostly to addition and multiplication but can also be applied to divisionin some cases.

Mathematically we define commutativity as:

The Distributive Property allows for multiplication across parentheses, and also quick multiplication of large numbers

by breaking them into sums of smaller numbers which are then easier to multiply.

For example:

Observe from the above equations that the term outside the parentheses distributes across all the terms inside the

parentheses.

Operators inside the parentheses must be addition or subtraction only and operators outside the parentheses must

be multiplication or division only for the distributive property to apply.

One common mistake is notdistributing the outside term across all the terms inside the parentheses like in the

example shown below

Another important fact to remember is that signs (+, -) also distribute equally across parentheses as in the following

example:

and


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You might notice that the above equation is the same as

The distributive property can also be applied to division:

Another interesting case is shown below:

This is the same as:

Solving One Variable Equations

A variable in mathematics is a symbol that is used in place of a value. The value of a variable depends on the other

values in the expression or equation. As a result, the variable can at times change accordingly.

Variables can be letters, Greek symbols or combinations of many other symbols.


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Expressions

An expression in mathematics refers to a combination of mathematical symbols and operators and operators. For

example,

is an expression since it combines the symbols 5 and 3 (in this case the symbols are real numbers) and the

mathematical addition operator (+).

An expression can also consist of variables as in the example below

wherex is a variable because it represents a value which we do no know at the moment.

Expressions can also be written without operators as long as variables are involved, for example, the following are

also expressions:

where t,x,y and are variables.

Equations

When you include an equals sign (=) in a mathematical expression, you end up with an equation. On either side of

the equals sign is an expression which leads to the definition of an equation as a mathematical statement that

asserts the equality of two expressions. In simpler terms, an equation is a statement which tells us that one thing is

equal to another.

Here are some examples of equations:

Solving Equations


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Now that we have established that an equation is a statement of equality, we're able to solve for unknown variables

in equations. Solving equations is a fundamental theorem of Algebra and Mathematics as a whole since all the

different aspects incorporate some sort of solving equations.

Example 1

Given the equation below, solve for the unknown variable:

Step 1

In the above, we currently aren't aware of what value the variable x represents, and so our task to find out what

that is.

The first step is to check how many variables we have and how many known values we have. In this example weonly have one variablex and two known values 3 and 6.

Step 2

Since all the variables are on the left hand side of the equals sign, let's focus on the expression on that side for a

minute,

We're adding some numberx to 3. If we look at the right hand side expression, 6, we realize that we need to find

some numberx to which you add 3 to get 6.

Step 3

From Elementary Algebra, we know that if x + 3 = 6, then we collect like terms and shift the numbers to one side

and leave the variable on the other side. To achieve this in this particular example, we subtract 3 from both sides of

the equation as below:

therefore,

Example 2

Solve fory in the following equation:


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Step 1

This example isn't so different from the previous one. The expression on the left containing a variable is equal to the

expression on the right.

Thus in this example we're finding a number represented by the variabley which when doubled and then 4

subtracted from it, will equal to 12.

Step 2

The first step in solving the above is to collect like terms, we need to put all the numbers on one side and leave only

the variable on the other side. We need to exercise caution here since, there is multiplication involved.

By adding 4 to each side, we get rid of it from the left hand side of the equation

Step 3

So now only 2 is left on the left hand side. We simply can't add or subtract to move it to the right hand side of the

equation since it is multiplied toy. To get rid of it, we pide through by 2 as below:

and we end up with

Thereforey = 8 is our solution.


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Matho Elementary Matho Prealgebrao Algebra

Basic Algebra Properties Associative Property Order of Operations Distributive Property Solving Equations Graphing Linear Equations Inequalities Graphing Linear Inequalities Slope Functions

FOIL Method Polynomials Factorization

Quadratic Equations The Quadratic Formula Complex Numbers Factor Theorem Remainder Theorem

Exponents Logarithms Square Roots and Radicals Rationalization Rational Expressions Conic Sections

Algebra Tutorso Geometry

o Trigonometryo Precalculuso Calculuso Statistics and Probabilityo Math Tutors

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Inequalities


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Solving inequalities is not so different from solving regular equations. In fact, an inequality sign (,,) is treated

the same as an equal (=) sign when solving inequalities involving only addition or subtraction. Before all that, let us

define the different inequality signs:

< indicates that the expression on the left is less than the expression on the right, forexample:

indicates that 3 is less than 9 and we all know that to be true.

means that the solution to the expression on the left is less than zero, but more on that

later.

> indicates that the expression on the left is greater than the expression on the right,for example:

shows that 9 is greater than 3 and similarly

means that the value of2x - 9 is greater than 11

indicates that the value on the left is less than or equal to the value on the right, forexample:

means that the value of the expression on the left must be less than or equal to 25

indicates that the value on the left is greater than or equal to the value on theright, for example:

Inequalities are best thought of as representing different regions on a number line:

< represents the region to the left of a given number, for example


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represents all the numbers to the left of 3 (less than 3), which in on the number line

below is shown by all the numbers in the direction of the arrow

> represents the region to the right of a given number, for example

represents all the numbers to the right of 3 (greater than 3), which in on the number line

below is shown by all the numbers in the direction of the arrow

represents the region from a given number to the left of that same number, forexample

represents all the numbers to the left of 5 (less than 5) including 5 itself as shown on the

number line below

represents the region from a given number to the right of that same number, forexample;

represents all the numbers to the left of 1 (less than 1) including 1 itself as shown on the

number line below


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Solving Inequalities

Most linear inequalities can be solved just the same as linear equations: Addition and subtraction of any number

(positive or negative) can be done to the expression on either side of the inequality without changing the inequality

itself. In other words, it would be the same as in any ordinary equation.

For example; solve for x in

Solution:

Subtract 9 from both sides of the inequality as follows:

and the answer would be

Observe that the above is solved in the same was as solving ordinary equations. This is because addition and

subtraction have no effect on the inequality sign. Multiplication and division are different, however, as the inequality

sign is treated differently depending on whether you are multiplying by a positive or negative number.

Multiplying or dividing through the inequality expression by a positive number has no effect on the inequality sign

and is treated as you would an ordinary equation.

For example, solve for x in:

Solution:

First add 9 to both sides of the inequality


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then divide through by 3

which results in

Multiplying or dividing through the inequality by a negative number has the effect of reversing the inequality sign, for

example from < to > as shown below

is solved as follows;

dividing by -2 reverses the inequality sign resulting in:

To prove why the above is true, let us first understand the answer:

means thatx can take on any value as long as that value is greater than -4 and the original equation will be true. To

prove this, let us try different values of x;

First, try x = 1, 1 is greater than -4 so substituting x = 1 in the original expression should give a mathematically

correct inequality

which is true.


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Next, let us try a value of x less than -4, for example, substitute x = -5

but 7 is not less than 5 which means that the solution we got as x > -4 is true. Try substituting different values of x

into the expression -3 - 2x < 5 and no matter what value you choose, as long asx > -4 the solution should always

hold true.

Solving Polynomial Inequalities

Solving more complicated polynomial inequalities is not so straightforward.

For example, solve the inequality below for x

Solution:

If the above had been an equation, finding the roots by factoring or completing the square would be all thats

necessary. However, inequalities are different. The above is solved as follows:

First factor the expression on the left

which means that either the solution to the expression on the left is x = -2 orx = 6 but this is not the end.

Next you have to test the different regions on the number line to find out exactly where the solution to the entire

inequality lies. First testx < -2 by picking a number on the left of-2 on the number line and then substitute it into

the original inequality i.e.


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which is not true and so we conclude that x is not less than -2.

Next we testx > -2 and by picking a number on the right of -2 on the number line, but for the time being this

number has to be less than 6

which is true and so we conclude thatx > -2. But since we had 2 roots, we have to testx = 6 as well.

We don't have to test forx < 6 since any number less than 6 is also greater than -2 and we already proved thatx >

-2. So we test forx > 6

which is not true, 9 is not less than zero and so the x is not greater than 6. This implies that x < 6

And thus the solution tox^2 - 4x - 12 < 0 can be given as:

which means that x lies on the region on the number line between -2 and +6


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FOIL MethodThe FOIL method is an important algebra method that defines how two binomials are multiplied. A binomial is

a polynomialwith two terms; and a polynomial is an expression of finite length where the variable is only affected by

addition, subtraction and multiplication but not division.

An example of a polynomial is given below

A binomial expression is of the following form

where each set of parentheses is a binomial. Thus a binomial is a can be factored into a product of two binomials.

Each term (each letter) is referred to as a monomial. The FOIL method is a standard algorithm for solving binomials

of the form above.

FOIL is an acronym for:

First - first multiply the first term in each set of the parentheses

Outside - then multiply the outside term in each set of the parentheses

Inside - then multiply the inside term in each set of the parentheses

Last - lastly multiply the last term in each set of the parentheses

Applying the FOIL method to the expression

would result in the following

where:

acis the product of the First terms in each set of parentheses


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adis the product of the Outside terms in each set of parentheses

bcis the product of the Inside terms in each set of parentheses

bdis the product of the Last terms in each set of parentheses

The FOIL method is the same as a two-step distributive property method and the above algorithm can be thought of

as

which turns out as

Examples of the FOIL Method

Solve the following expressions

Example 1

Step 1

Step 2

Example 2

Step 1

Step 2

Step 3

Example 3


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Step 1

Step 2

Example 4

Step 1

Example 5

Step 1

Step 2

Example 6

Step 1

Step 2

Step 3


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Polynomials

A polynomial is defined as an expression of finite length consisting of variables with only positive whole numberexponents. For example:

is a polynomial of degree 2.

The degree of a polynomial is the exponent of the leading term of the polynomial. In the polynomial above, each of

the following refers to one term; is one term; 4x is another term; and 4 is the last term. The leading term is the

term whose exponent has the highest value and is often the first term of the polynomial. In the polynomial above,

the leading term is . A term is also referred to as a monomial; which means a polynomial of one term. A term

consists of a variable (in this casex) and a constant or coefficient (any number). This is true despite there being no

obvious variable attached to the last term as shown below:

Keeping in mind that any number to the power zero is one (i.e. = 1), the above becomes:

Polynomials are commonly denoted by P(x), for example

Roots of Polynomials

Equating P(x) to zero and solving forx is referred to as the solving for the roots ofP(x). A root of a

polynomial P(x) is defined as the value ofx for which the polynomial equals zero. Roots of polynomials can be either

positive or negative or zero. Roots can also be real numbers or complex (imaginary) numbers. If a polynomial has

degree n, then there must exist n roots of that polynomial. If you were to graph a given polynomial, the roots of the

polynomial would be the points where the curve formed by the polynomial cuts the x-axis (the x-intercepts).

Given the polynomial below

from the degree of the polynomial (3) we can tell that the polynomial will have 3 roots such that


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If the roots of the above polynomial are , and , then these roots are related by

where each expression within a set of parentheses is known as a factor of the polynomial (i.e. (x - ) is a factor of

the polynomial).

Roots of the polynomial P(x) can be found by equating the polynomial to zero and then trying to substitute for

different values ofx to find out which ones make the equation equal to zero. This is known as factoring.

For example; find the roots of the polynomial

Step 1

First step is to equate the polynomial to zero

Step 2

next step is to try different values ofx to find 3 that make the equation equal to zero

try substitutingx = 1

this implies thatx = 1 is a root of the polynomial and (x - 1) is a factor of the polynomial

but the degree of the given polynomial is 3 so we need to find 2 other roots to make a total of 3 roots

Step 3

try substitutingx = -1

this implies that x = -1 is not a root of the polynomial

Step 4


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try substituting x = 2

this implies thatx = 2 is a root of the polynomial and (x - 2) is a factor of the polynomial

Step 5

try substitutingx = 3

Step 6

Therefore the roots of the polynomial are given by x = {1,2,3} and this can be proved by expanding the factors, i.e.

which is the same polynomial we started with!

If we look at the graph of this function, we can clearly see the roots of our function.


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Quadratic Polynomials

Quadratic Polynomials are polynomials of degree 2; meaning that the leading term has a variable with an exponent

of two. Quadratic polynomials are very important in lower level algebra because of the special properties they

possess, and thus warrant an entire section of their own under quadratic equations.

FactorizationFactorization is defined as the process of breaking down a number or an expression into a product of different

numbers or expressions called factors. In other words, factorization refers to breaking down large and at times

complicated expressions into a product of smaller ones that are then easier to deal with. You can also think of

factorization as the opposite ofdistribution.

For example, can be factored into by extracting the common factor of 3x.

Factorization can be done in three ways:

by taking the difference of two squares

by grouping and

by breaking up the expression into perfect trinomials.

Factorization by Taking Difference of Two Squares

The difference of two squares is also known as the difference of perfect squares and refers to the mathematical

identity


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The above identity is true regardless of the the coefficients of a and b, given that the coefficients are the same for a

and b.

This is because expanding (a + b)(a - b) results in similar terms that cancel each other out, i.e.

ab and ba are the same and thus can be added or subtracted, and in this case their difference is zero, which leaves

us with

Certain polynomials can be factored using the difference of two squares, but these polynomials must have squares in

them. For example

can be factored using the difference of two squares into

Similarly

can be factored using the same method into

This can be further factored using the same method into

By equating the polynomial to zero, the roots of the polynomial can be easily found from the factors.

Factorization by Grouping

Factorization by grouping involves separating the given expression into smaller groups and then factoring those

groups independently. The tricky part about this becomes choosing which expressions to group together. It wouldn't

make much sense to group expressions with odd coefficients with those with even coefficients but there is no

consensus on how to pick the different groups. It's up to you to decide what makes sense, and the more practice you

have with grouping, the more obvious it becomes.


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For example, factor

solution

The first step is to pick which expressions to group together. In this case it should be obvious from the coefficients.

Observe that the operator between the different sets of parentheses is an addition operator (+). This is important to

remember because only sum / difference operators are allowed.

Factoring the separate groups results in

The above is done by extracting the common factors to each set of parentheses.

Now observe that (x + 3) is a common factor to both expressions and can be factored out as:

Its important to remember that you should NEVER divide through by anything since you might lose some factors by

doing so.

Factorization into Perfect Square Trinomials

Perfect square trinomials refer to quadratic polynomials which can be factored into identical binomial expressions.

These quadratic polynomials are of the standard form:

and

For more on factoring quadratic polynomials, refer to section on quadratic equations.

Complex Numbers


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In algebra, there are two types of numbers: real numbers and imaginary numbers. Real numbers refer to any

ordinary number (e.g. 1, 2, 3 . . .) while imaginary numbers are . . . well . . . imaginary! They don't really exist, they

are represented by a real number with the letter i next to it. For example, 3i is an imaginary number.

Complex numbers are those consisting of a real part and an imaginary part, i.e.

where ais the real part and bi is the imaginary part.

Imaginary numbers are called so because they lie in the imaginary plane, they arise from taking square roots of

negative numbers. The i on an imaginary number is equal to the square root of negative one, i.e.

The i was introduced in order to simplify the problem of taking square roots of negative numbers. For example, if we

can find the square root of negative nine as follows:

using properties of square roots, the above becomes

and since is represented by i

Properties of Imaginary NumbersAddition

Imaginary numbers behave like ordinary numbers when it comes to addition and subtraction:

Multiplication

From the section on square roots, you should know that the following is true:

Therefore, it should follow that the following should also be true:


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since i = -1, and

Exponents

For any even number n, the following is always true

if an only if the following is also true

For example, given n = 4, an even number:

Conversely, if is an odd number, then the following is true:

For example; given where n = 6

then

For any odd number m greater than 1, the following is always true:

Whether i is positive or negative depends on the value of m. When working with with odd number powers of i, you

always split the powers into a sum of even and odd numbers. For example:


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which is the same as

The even part of the exponent determines whether i is positive or negative as determined in the previous property.

Division

Imaginary numbers can be divided just as any other number if there is only one term:

If there are two terms divided by two terms, we use the complex conjugate

To evaluate the following complex number, we multiply by the complex conjugate over itself.

We multiply by the complex conjugate of the denominator to eliminate the complex number and make it a real

constant.

As mentioned earlier, complex numbers consist of both a real and an imaginary part. Any imaginary number can also

be considered as a complex number with the real part as zero, i.e.


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It is important to remember that the real and imaginary parts of the complex number do no interact directly, for

example:

When adding or subtracting complex numbers, add the real part to the real part and the imaginary part to the

imaginary part:

Multiplication and division can be done on a complex number using either a real or imaginary number, i.e.

It is important to remember that when writing a complex or imaginary number, do not write the imaginary part in the

denominator like this:

In such situations, we rationalize the denominator to become:

For more on rationalization, refer to the section on rationalization.

Complex Numbers ExamplesExample 1

Solve the following

Step 1

Step 2


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Step 3

remember that i x i = -1

Step 4

Example 2

Evaluate the following:

Step 1

This example serves to emphasize the importance of exponents on i. The first step is to inspect all the exponents and

apply the properties we listed above.

Step 2

Step 3

Step 4

putting it all together


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Remainder Theorem

Before tackling Remainder theorem, you might want to revisit long division (also known as synthetic division) and quadratic

equations. Just like numbers, polynomials can be divided through by numbers.

The Remainder Theorem is a useful mathematical theorem that can be used to factorize polynomials of any degree in a neat and

fast manner.

The Remainder Theorem states that when you divide a polynomial P(x) by any factor (x - a); which is not necessarily a factor of

the polynomial; you'll obtain a new smaller polynomial and a remainder, and this remainder is the value ofP(x) at x = a, i.eP(a)

Remainder Theorem operates on the fact that a polynomial is completely divisible once by its factor to obtain a smaller

polynomial and a remainder of zero. This provides an easy way to test whether a value a is a root of the polynomial P(x).

For example, given a polynomial P(x), and also given that a is a root of the polynomial, then when P(x) is divided by the factor(x

- a), the result should be a smaller polynomial P1(x) and a remainder zero.

Below is an example that serves to prove the remainder theorem

prove that x = 1 is a root ofP(x),

solution:

which implies that x = 1 is a root of the polynomial P(x), and (x - 1) is a factor ofP(x)

Therefore if we were to synthetically divide through P(x) by (x - 1), we should get a new smaller polynomial and a remainder of

zero:

Remainder Theorem Examples


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The first step of syntheic division is to arrange the polynomial and the factor in the format below. The factor is the divisor and is

on the outside, while the polynomial is the dividend and goes under the division bar.

The next step is to divide the first term of the polynomial (the first term should be the one with the highest power) by the x part of

the factor. In this case, you divide x3 by x, to get x2 which you then write at the top of the division bar.

Next you multiply the term that you wrote at the top by the entire divisor, in this case you multiply x2 by (x - 1) to get (x3 -

x2)which you then subtract from the polynomial under the division bar as shown below.

By subtracting, you get rid of the term with the largest exponent to reduce the size of the polynomial which is a remainder as

shown below:

Now we have a new polynomial for which we repeat the second step. In this example, we divide -5x2 by x and add the result

which is -5xto the term already at the top of the division bar

Next we repeat the third step, we multiply this result which is on top of the division bar (only the most recently added) by the

entire divisor and then subtract what you get from the divident polynomial, as shown in this step below:


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Once again we have a new polynomial under the division bar.

As you have probably guessed, we repeat the second step by dividing the first term of the polynomial by the x part of the divisor

and add the result to whatever is already on top of the division bar.

Next we multiply the most recently added term by the entire divisor and again subtract the result from the dividend polynomial.

Now the subtraction step is repeated to obtain a new polynomial.


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The remainder here is zero because we have completely divided through P(x) by the factor (x - 1). When you reach this step,

you're done, dividing any further by zero would result in zero.

Now lets try to use the remainder theorem to find a value ofP(-1), to see what we result with. We already know that x = -1 is not

a root of the polynomial P(x)

The first step is the same as in the previous example; arrange the divisor and dividend on the outside and inside of the division

bar respectively.

Next we perform the first division; the first term of the dividend by the x term of the divisor

Then comes the multiplication; the most recent result by the entire divisor and subtract the result from the dividend

The subtraction gets rid of the term with the largest power and yields a new dividend under the division bar.

We once again divide the first term of the dividend by the x term of the divisor and add the result to whatever is already on top of

the division bar.


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Next we repeat the multiplication step

Then we subtract the result from the dividend to obtain a new polynomial dividend.

Once again we divide the first term of the dividend by the x term of the divisor.

Multiplication once again:


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Then subtraction:

and we're left with a remainder. We can't continue division any further since the x term in the divisor has a higher exponent (x1)

than the x in the dividend polynomial (which in this case happens to be x0) and we don't want to end up with negative exponents

so we stop there and say that whatever is left is the remainder which in this case is -24.

Next we check to see if the value ofP(-1) is the same as the remainder obtained above

Thus we have proved the remainder theorem. You may try a few more values ofx as a way of practicing synthetic division of

polynomials.

Solving Polynomials Using the Remainder Theorem

From the section on polynomials, we know that the root of a polynomial P(x) is defined as the value ofx for which the

polynomial is equal to zero.

Using Remainder Theorem, we can redefine a root as a value a for which the factor (x - a) divides through the polynomial P(x)to

get a remainder of zero. In other words it divides through the polynomial completely.

The remainder theorem is especially useful in finding the roots of polynomials of a large degree (4 or more). By finding the one

root and using it to synthetically divide through the polynomial, we're able to obtain a smaller polynomial for which we repeat

the process until we find the last root.

The example below better explains the steps involved;

Given the polynomial P(x) below, find all its roots.


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Solution:

From the degree of the polynomial P(x) which is 5, we can tell that P(x) will have 5 roots.

The first step is to use small numbers values ofx to find which ones will give a value of zero ie we need to find

So we first attempt x = 0, we substitute for x = 0 into P(x)

since P(0) is not equal to zero, we can safely conclude that 0 is not a root ofP(x)

Next we move on to another number, let's try substituting for x = -3

P(-3) = 0 which means that -3 is a root of the polynomial and that (x + 3) is a factor of the polynomial. So now we can use

synthetic division to get a smaller polynomial

We obtain a new polynomial, lets call this polynomial f(x)

We then repeat the process for f(x). We try to find a root off(x) by trial and error

First we try x = 1


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f(1) is not equal to zero, so we move on to another value ofx

Try x = 3

f(3) = 0 which implies that x = 3 is a root of the polynomial and (x-3) is a factor.

Next we divide f(x) by (x - 3)

We obtain a new polynomial g(x) and a remainder of zero

We repeat the process one last time.

Since we can see that g(x) is a polynomial of degree 2 and that there only addition operators in the polynomial, we can conclude

the that one of the roots ofg(x) must be a negative.

So we try to substitute for x = -4

since g(4) is equal to zero, we conclude that x = -4 is a root of the polynomial and (x+4) is a factor, which allows us to perform

synthetic division on g(x)


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from which we get the last factor of the P(x) as (x + 3) and thus the root as x = -3.

So now we can rewrite P(x) as

and the roots ofP(x) are x = {4,-3,-3,3}

>

Exponents

Exponents are algebraic operators that are used to multiply a number by itself a certain number of times. Exponents

are also known as powers of numbers. Exponents are written as

where a is referred to as the base and n is known as the exponent, and the whole expression is said to be: a raised

to the power ofn.

is the same as saying: multiply a by itselfn times. i.e.

where the last term is the nth term.

An exponent can be positive or negative, whole or fractional, a value or a variable and all these cases are dealt with

differently.

Positive Exponents

Positive Exponents or powers have the effect of multiplying the base by itself as many times as the value of the

exponent. As a general form, we say that if n is positive, then


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is the same as multiplying a by a, n times.

For example

is the same as multiplying 3 by itself 4 times, i.e.

Raising a number or variable by the power of two is in effect squaring the number.

Similarly, raising a number or variable to the power of three is in effect cubing the number.

Negative Exponents

Negative exponents or powers have the effect of reciprocating the number or variable on which they appear, i.e.

observe that the denominator now contains the base raised to the positive power. In other words, numbers raised to

negative exponents are the same as finding the reciprocal or inverse of the same number raised to the positive base.

(Reciprocal means 1 divided by the number).

For example

The only exception to the rule above is if the base is zero. Remember that dividing by zero is not allowed, and zero

raised to a negative power would result in 1 divide by zero:


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which is not allowed.

Special Exponents

Certain numbers as exponents cause the base to behave in a special way.

Zero Exponent

As a general rule, any variable or non-zero number raised to the power zero is equal to one.

This is always true regardless of what the base a is. The base can be a non-zero number or a variable, positive or

negative and, as long as it is raised to the power zero, it will equal to one. The only exception is the number zero

itself because when zero is raised to the power zero, the result is still zero.

So it is important to remember that any other number, except zero, when raised to the power of zero is equal to one.

For example:

Positive One Exponent

The number one is another interesting exponent. Any number raised to the power of positive one is equal to the

number itself. This is regardless of whether the number is positive or negative.

and

Odd and Even Whole Number Exponents

When a positive number is raised to an odd or even whole number, the result is always positive. Negative numbers,

on the other hand, behave differently.


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When a negative number is raised to an even whole number power, the result will ALWAYS be a positive number. For

example:

The above is also true if the exponent is a negative whole number:

You can try out different negative numbers and raise them to even whole numbers and the result is always a positive

number.

Conversely, when a negative base is raised to an odd whole number exponent, the result is ALWAYS a negative

number. For example:

the same applies to negative exponents:

Exponents of Ten

Ten (10) is a special number because raising ten to any whole number exponent is the in effect adding a number of

trailing zeros to 10, and these are as many as the value of the exponent, i.e.

where the last 0 is the nth zero.

For example:

Fraction Exponents

Exponents can be whole numbers or fractions. Whole number exponents have been discussed above. Fractional

Exponents behave different, instead of having the effect of multiplication of the base by itself, they have the effect of

finding the root of the base. i.e.


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is the same as finding the nth root of a, which is also written as:

Thus to find the square root of a number, we can also write:

n can be positive or negative, but in order to get real roots, a must be positive. Negative a would result in complex

roots. For example;

Properties of Exponents

Numbers or variables with exponents satisfy the following properties which makes them easy to manipulate.

If the bases in a given exponential equation are equal, the exponents are also equal.

i.e.

would imply that

More on this later.

When multiplying two numbers or variables with the same base, the effect is the same as

adding their exponents.

i.e.

The above is true regardless of what the base is or what values the exponents have. For

example:

This can be proved as shown below:


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When dividing two numbers or variables with the same base, the effect is the same as

subtracting their exponents from each other as shown below:

Observe that the exponent of the denominator is subtracted from the exponent of the

numerator.

For Example:

Which we can prove as follows:

Exponents distribute equally into parentheses as shown below

Exponents can only be multiplied or divided if they are in the following form:

Exponential operations are NOT associative i.e.

When solving expressions such as the one above, follow the procedure as in the example

below:

First only consider the exponent of 4


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then substitute the answer back

Solving Exponential Equations

Exponential equations are those with polynomials as exponents, for example:

Ordinarily, such exponential equations would be solved by using logarithms (refer to section on logarithms), but

some exponential equations can be solved using the property mentioned above

applying this property to the equations above:

would become

which then implies thatx = 2 since the bases are the same

Similarly,

is the same as

from which you would proceed to solve for x.

From the above example, you should have noticed that we had to put all the components of the exponential equation

into the same base in order to apply that property.

So this property will only apply if the bases can be expressed in one base. For example, while the property can be

used to solve the equation below


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because 128 and 16 can be expressed as exponential functions of the same base, i.e.

which using another of the properties mentioned before becomes

which then leads to:

from which you can solve for x (refer to section on polynomials)

The same can not be said of the exponential equation below:

This is because 2 and 81 can't be expressed into exponential equations of the same base. This kind of exponential

equation can however be solved using logarithms.

A sure way to tell if the bases can be expressed into exponential equations of the same base is to check if they are

multiples of each other. In other words, check if they share a common factor that factors both of them completely

leaving no remainder.

In the examples above, 128 and 16 are multiples of each other and share 2 as a common factor. In the second

example, 2 and 81 are not multiples of each other and share not factors.

For a more in depth explanation on exponential functions, see exponential functions in Precalculus.

Square Roots and Radicals

A square root is defined as a number which when multiplied by itself gives a real non-negative number called a

square.

A square root is best defined using geometry where, considering a square (which is a four sided polygon whose sides

are all equal), a square root is defined as the length of the diagonal of this square (a diagonal is a line drawn from

one vertex/corner to the opposite vertex of the square).


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A radical is a root of a number. A square root is a radical. Roots can be square roots, cube roots, fourth roots and so

on.

A square root is commonly shown as

where is known as the radical sign and is known as the radicand.

A square root of a number can also be represented as

and a radical as

where we say that in the above, we're finding the nth root ofx. For more on the above notation, refer to section

on exponents.

A radical can also be represented as

A square root is also represented as

A cube root as

A fourth root as

Every square has two square roots; one positive and the other negative. This is shown as:

which is written as


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This can be proved in the following way. Consider a number, a

but also

the latter is because a negative multiplied by a negative equals a positive.

And so it follows that

For example,

but also

Therefore,

Thus it follows that any real positive number has two roots. But when talking about radicals

in other words, only refers to +x which is known as the principal square root. So despite having said above

that

we usually only consider

especially if the is used.

But if the question asked is in the form

always give both the positive and negative roots, i.e.


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Although any real positive number can be considered a square number and thus has a square root, we only consider

numbers with whole number square roots as squares.

For example

Properties of Square Roots and Radicals

Properties of square roots and radicals guide us on how to deal with roots when they appear in algebra.

Examples of Square Roots and Radicals

Evaluate the following:

1.

Solution:

2.

Solution:


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3.

Solution:

4.

Solution:

The above is left as is, unless you are specifically asked to approximate, then you use a calculator.

5.

Solution:

Rationalization

Rationalization, as the name suggests, is the process of making fractions rational. The need for rationalization arises

when there are irrational numbers, surds or roots (represented by ) or complex numbers in the denominator of a

fraction.

The following are examples of fractions that need to be rationalized:


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Although fractions with surds and complex numbers in their denominators are not wrong, it is a general consensus

that they should not be written that way hence the need to simplify them by rationalization.

Rationalization is all about moving the surd ( ) or complex number to the numerator. Rationalization does not

change the value of a number or function but only re-writes it in a more acceptable and most times easier to

understand form.

Rationalization of fractions involves the use of conjugates.

You should observe from above that a conjugate is formed by changing the sign in front of either the surd or the

complex number. This is not a rule but it is a good practice for the sake of uniformity.

Conjugates are useful because they when a number is multiplied by its conjugate, the result will not have surds or

complex numbers in it.

Rationalization of Surds

As mentioned above, leaving surds in the denominator of a fraction is not good mathematical practice. So the

question becomes how to get rid of them without changing the actual number or function.

The answer to this is simple: Multiply the surd in the denominator by its conjugate to get rid of the surd. This works

because


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But in order not to change the number, you multiply the denominator and the numerator by the same conjugate

which is in effect multiplying by one.

and it should be clear that

So the above becomes

To prove this, let's use a value ofaas 4

therefore if asked to solve for following

This example is relatively simple enough to be a good proof that rationalization does not change a number

This is the same answer you would get if you solved directly as


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A better example of when this would be more useful is solving the following:

This is solved the same as the previous example:

considering only the denominator;

substituting the above into the fraction;

The above algorithm works with any real values ofaand b, as in the example below:

Since there is no obvious way to simplify the above without a calculator, we rationalize it:


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As you can see, the above is much easier to understand than the original expression. If you want to confirm that

they are indeed the same, use a calculator to compare the values.

One common mistake most students make when rationalizing fractions is to misplace the signs on the conjugate, for

example

is correct, while

and

are both wrong because they change the value of the expression. Therefore, it is important to remember to be

consistent and not misplace any signs as that changes the entire expression.

One sure way to check if you have made any errors is to observe whether or not the denominator factors as nicely as

in the examples before. If it does not, check your work for errors.

Rationalization of Complex Numbers

Complex numbers with imaginary numbers in the denominator are rationalized in a similar manner to the procedure

outlined above. The identity that allows for rationalization of complex numbers is shown below:

because


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and from the section above we've seen that

therefore

As in surds, in order not to change the number when rationalizing, we multiply both the numerator and denominator

by the same conjugate which is in effect multiplying by one, i.e.

The conjugate used in rationalizing complex numbers is called a Complex Conjugate because the imaginary part of

the complex number is the one that gets conjugated (as in the complex conjugate of (a + bi) is (a - bi)).

Rationalization of complex numbers always follows the following algorithm:

Given a complex fraction of the form


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To further illustrate the algorithm above, let's take the following example:

Rationalize the fraction below

As always, begin by multiplying both the numerator and denominator by the complex conjugate of the denominator.

Rational Expressions

A rational expression, also known as a rational function, is any expression or function which includes a polynomial in

its numerator and denominator. In other words, a rational expression is one which contains fractions of polynomials.

For example:


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The last equation also has a polynomial in the denominator, keeping in mind that thus

becomes

The important thing to remember is that the denominator must never equal to zero, otherwise you'll end up dividing

by zero.

When asked to find the domain of a rational function, though solving may result in many variables, you must always

pick only those which will result in the polynomial in the denominator not equal to zero.

Rational Expression Examples

For example; find the domain of

What the question is asking for are the values of x for which the rational function is said to exist or make

mathematical sense. In other words, find the values of x for which the denominator is not equal to zero. So the firststep is equating the denominator to zero i.e.

from which you can see that


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and then we say that the domain is: all values of x except for x = 3

Notice on the graph of the function, we have an asymptote atx = 3 which means that this value is not in the

domain. If it is not in the domain, then a range value (y-value) cannot exist.

Example: Find the domain of the expression below

As before, start with equating the denominator to zero and then find factor the resulting equation to find its roots

which means that the roots of the denominator are


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These are the values for which the denominator is equal to zero, thus we say that the domain of the expression is

given by:

all values of x except

Example: Find the domain of

Equate the denominator and factor


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so the whole rational expression becomes

Although we have expressions in both the denominator and denominator, the expression in the numerator does not

affect the domain of the entire rational expression, so we only consider the denominator

Therefore,

which means thatx = {1,3,4}

And thus the domain of the rational expression is:

all values of x except forx = {1,3,4}

Simplifying Rational Expressions

Rational Expressions can be factored and simplified as in the example below:

First factor both numerator and denominator


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then you can see that x is a common factor in both the numerator and denominator, so the above is the same as:

However, it is important to remember you should never simplify the rational expression before finding the domain. In

case you still feel like simplifying before finding the domain, then you must keep track of the factors which you

'cancel' out.

For the example above, to find the domain from the simplified expression

set the denominator equal to zero, then solve for x

from which

However,x = 2/3 is not the only factor for which the denominator of3x/(2x - 3x2) is equal to zero. Since we

divided through by a factor to get the simplified expression, we must set that factor to zero as well and solve for x.

In this case since we divided through by x, we say

and then we give the domain as: all values of x except for x = {0,2/3}


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Example: Simplify the rational expression and the also state the domain

Step 1

First factor both the numerator and denominator to get

Step 2

In this form, it should be easy to see the common factors

Step 3

but (x - 3) and (3 - x) are very similar can can be manipulated so that we can also cancel them from the expression

Step 4

factoring out -1 gives


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Step 5

substituting the above into the expression

the above is the simplified expression needed.

Step 6

To find the domain; equate the factors to zero to get the points where the denominator will be zero i.e.

and the domain is given as:

all values of x except

rules of algebra

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