Polynomials: Definitions

By now, you should be familiar with variables and exponents, and you may have dealt with expressions like 3x4 or 6x. Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Here are some examples:

 6x –2 This is NOTa polynomial term...

...because the variable has a negative exponent.

 1/x2This is NOT

a polynomial term......because the variable is

in the denominator.

 sqrt(x) This is NOTa polynomial term...

...because the variable is inside a radical.

 4x2 This IS a polynomial term... ...because it obeys all the rules.

Here is a typical polynomial:

Notice the exponents on the terms. The first term has an exponent of 2; the second term has an "understood" exponent of 1; and the last term doesn't have any variable at all. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the largest exponent first, the next highest next, and so forth, until you get down to the plain old number.

Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you may put in for the variable x, that constant term will never change. In the picture above, no matter what x might be, 7 will always be just 7.

The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term".

The exponent on a term tells you the "degree" of the term. For instance, the leading term in the above polynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial". Here are a couple more examples:

Give the degree of the following polynomial:  2x5 – 5x3 – 10x + 9

This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.

This is a fifth-degree polynomial.

Give the degree of the following polynomial:  7x4 + 6x2 + x

This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.

This is a fourth-degree polynomial.

When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the "leading" coefficient.

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is 3; the constant term doesn't have a coefficient.   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

The "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names:

a one-term polynomial, such as 2x or 4x2, may also be called a "monomial" ("mono" meaning "one") a two-term polynomial, such as 2x + y or x2 – 4, may also be called a "binomial" ("bi" meaning "two") a three-term polynomial, such as 2x + y + z or x4 + 4x2 – 4, may also be called a "trinomial" ("tri" meaning "three")

I don't know if there are names for polynomials with a greater numbers of terms; I've never heard of any names other than what I've listed.

Polynomials are also sometimes named for their degree:

a second-degree polynomial, such as 4x2, x2 – 9, or ax2 + bx + c, is also called a "quadratic" a third-degree polynomial, such as –6x3 or x3 – 27, is also called a "cubic" a fourth-degree polynomial, such as x4 or 2x4 – 3x2 + 9, is sometimes called a "quartic" a fifth-degree polynomial, such as 2x5 or x5 – 4x3 – x + 7, is sometimes called a "quintic"

There are names for some of the higher degrees, but I've never heard of any names being used other than the ones I've listed.

By the way, yes, "quad" generally refers to "four", as when an ATV is referred to as a "quad bike". For polynomials, however, the "quad" from "quadratic" is derived from the Latin for "making square". As in, if you multiply length by width (of, say, a room) to find the area in "square" units, the units will be raised to the second power. The area of a room that is 6 meters by 8 meters is 48 m2. So the "quad" refers to the four corners of a square, from the geometrical origins of parabolas and early polynomials.

Rational ExpressionsA 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:

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 first step is equating the denominator to zero i.e.

from which you can see that

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 at x = 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

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

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 that x = {1,3,4}

And thus the domain of the rational expression is:

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

Simplifying Rational Expressions

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

First factor both numerator and denominator

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 of 3x/(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}

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

Periodic FunctionIn mathematics, a periodic function is a function that repeats its values in regular intervals or 

periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibitperiodicity. Any function which is not periodic is called aperiodic.

Definition

A function f is said to be periodic if

for all values of x. The least positive constant P with this property is called the period. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods.

Geometrically, a periodic function can be defined as a function whose graph exhibitstranslational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

A function that is not periodic is called aperiodic.

Even and Odd FunctionsThey are special types of functions

Even Functions

A function is "even" when:

f(x) = f(-x) for all x

In other words there is symmetry about the y-axis (like a reflection):

This is the curve f(x) = x2+1

They got called "even" functions because the functions x2, x4, x6, x8, etc behave like that, but there are other functions that behave like that too, such as cos(x):

Cosine function: f(x) = cos(x)It is an even function 

But an even exponent does not always make an even function, for example (x+1)2 is not an even function.

 

Odd Functions

A function is "odd" when:

-f(x) = f(-x) for all x

Note the minus in front of f: -f(x).

And we get origin symmetry:

This is the curve f(x) = x3-x

They got called "odd" because the functions x, x3, x5, x7, etc behave like that, but there are other functions that behave like that, too, such as sin(x):

Sine function: f(x) = sin(x)It is an odd function

But an odd exponent does not always make an odd function, for example x3+1 is not an odd function.

Neither Odd nor Even

Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not have to be even or odd.

In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:

This is the curve f(x) = x3-x+1

It is not an odd function, and it is not an even function either.It is neither odd nor even!

Even or Odd?Example: is f(x) = x/(x2-1) Even or Odd or neither?Let's see what happens when we substitute -x:

Put in "-x": f(-x)= (-x)/((-x)2-1)

Simplify: = -x/(x2-1)

= -f(x)

 So f(-x) = -f(x) and hence it is an Odd Function

Special Properties

Adding:

The sum of two even functions is even The sum of two odd functions is odd The sum of an even and odd function is neither even nor odd (unless one function is zero).

Multiplying:

The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd function.

ONTO FUNCTION AND ONE-TO-ONE FUNCTION

Remember that a function is a set of ordered pairs in which no two ordered pairs that have the same first component have different second components.  This means that given any x, there is only one y that can be paired with that x.

Onto Function

A function f from A to B is called onto if for all b in B there is an a in Asuch that f (a) = b.   All elements in B are used.

Such functions are referred to as surjective.

"Onto"(all elements in B are used)

NOT "Onto"(the 8 and 1 in Set B are not used)

By definition, to determine if a function is ONTO, you need to know information about both set A and B. When working in the coordinate plane, the sets A and B may both become the Real numbers,

stated as  .

EXAMPLE 1:  Is f (x) = 3x - 4 onto where  ?

This function (a straight line) isONTO.

As you progress along the line,every possible y-value is used.

In addition, this straight line also possesses the property that each x-value has one

uniquey-value that is not used by any other x-element.  This characteristic is referred to as

being one-to-one.

EXAMPLE 2:  Is g (x) = x² - 2 onto where  ?

This function (a parabola) is NOT ONTO.

Values less than -2 on the y-axis are never used.  Since possible y-values belong to the

set of ALL Real numbers, not ALL possibley-values are used.

In addition, this parabola also has y-values that are paired with more than one x-value,

such as (3, 7) and (-3, 7).This function will not be one-to-one.

EXAMPLE 3:  Is g (x) = x² - 2 onto where  ?If set B is redefined to be  , ALL of the possible y-values are now used, and  function g (x)  (under these conditions) is ONTO.

   

One-to-One Function

 A function f from A to B is called one-to-one (or 1-1) if whenever f (a) = f (b) then a = b.   No element of B is the image of more than

one element in A.

In a one-to-one function, given any y there is only one x that can be paired with the given y.  Such functions are referred to as injective.

"One-to-One" NOT "One-to-One"

 

EXAMPLE 1:  Is f (x) = x³  one-to-one where  ?

This function is One-to-One.This cubic function possesses the property

that each x-value has one unique y-value that is not used by any other x-element.  This characteristic is referred to as being 1-1.

Also, in this function, as you progress along the graph, every possible y-value is used,

making the function onto.

EXAMPLE 2:  Is g (x) = | x - 2 | one-to-one where  ?

This function is NOT One-to-One.

This absolute value function has y-values that are paired with more than one x-value, such

as (4, 2) and (0, 2).This function is not one-to-one.

In addition, values less than 0 on the y-axis are never used, making the function NOT

onto.

EXAMPLE 3:  Is g (x) = | x - 2 | one-to-one where  ?With set B redefined to be  , function g (x) will still be NOT one-to-one, but it will now be ONTO.

 

BOTH

Functions can be both one-to-one and onto. 

 Such functions are called bijective.Bijections are functions that are both injective and surjective.

"Both" NOT "Both" - not Onto

Examples of functions that are BOTH onto and one-to-onecan be seen in each of the categories above.

Inverse FunctionsAn inverse function goes in the opposite direction!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram:

The Inverse Function just goes the other way:

So the inverse of:   2x+3   is:   (y-3)/2

The inverse is usually shown by putting a little "-1" after the function name, like this:

f-1(y)

We say "f inverse of y"

So, the inverse of f(x) = 2x+3 is written:

f-1(y) = (y-3)/2

(I also used y instead of x to show that we are using a different value.)

Back to Where We Started

The cool thing about the inverse is that it should give you back the original value:

 If the function f turns the apple into a banana,

Then the inverse function f-1 turns the banana back to the apple

Example:Using the formulas from above, we can start with x=4:

f(4) = 2×4+3 = 11We can then use the inverse on the 11:

f-1(11) = (11-3)/2 = 4And we magically get 4 back again!We can write that in one line:

f-1( f(4) ) = 4"f inverse of   f of 4   equals 4"

So applying a function f and then its inverse f-1 gives us the original value back again:

f-1( f(x) ) = x

We could also have put the functions in the other order and it still works:

f( f-1(x) ) = x

Example:Start with:

f-1(11) = (11-3)/2 = 4And then:

f(4) = 2×4+3 = 11So we can say:

f( f-1(11) ) = 11"f of   f inverse of 11   equals 11"

 

Solve Using Algebra

You can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function: f(x) = 2x+3

Put "y" for "f(x)": y = 2x+3

Subtract 3 from both sides: y-3 = 2x

Divide both sides by 2: (y-3)/2 = x

Swap sides: x = (y-3)/2

Solution (put "f-1(y)" for "x") : f-1(y) = (y-3)/2

This method works well for more difficult inverses.

Fahrenheit to Celsius

A useful example is converting between Fahrenheit and Celsius:

To convert Fahrenheit to Celsius: f(F) = (F - 32) x 5/9

The Inverse Function (Celsius back to Fahrenheit) is: f-1(C) = (C × 9/5) + 32

For You: see if you can do the steps to create that inverse!

Inverses of Common Functions

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Here is a list to help you:

Inverses Careful!

<=>

<=> Don't divide by zero

<=> x and y not zero

<=> x and y ≥ 0

<=>  or n not zero

(different rules when n is odd, even, negative or positive)

<=> y > 0

<=> y and a > 0

<=> -π/2 to +π/2

<=> 0 to π

<=> -π/2 to +π/2

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

CONSTANT FUNCTIONA constant function is used in math to refer to a function whose value does not vary and thus making it a constant value. The function's value does not change even for the variables values as well as the parameters change.