elementary algebra exam 4 material exponential expressions & polynomials

Elementary Algebra

Exam 4 Material

Exponential Expressions & Polynomials

Exponential Expression

• An exponential expression is:

where is called the base and is called the exponent

• An exponent applies only to what it is immediately adjacent to (what it touches)

• Example:

nana

23x 3 not to x,only to appliesExponent 4m negative not to m, only to appliesExponent

32x (2x) toappliesExponent

Meaning of Exponent

• The meaning of an exponent depends on the type of number it is

• An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself

• Examples: 23x 4m

32x

xx3mmmm 1

xxx 222 38xexponentinteger any of meaning learn the willsection wenext In the

Rules of Exponents

• Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents

• Examples:

nmnm aaa

24 33 243 63 47 xx 47x 11x

Rules of Exponents

• Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents

• Examples:

mnnm aa

243 243 83

47x 47x

28x

Rules of Exponents

• Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power

• Examples:

nnn baab

23x 223 x 29x

42y 442 y 416y

Rules of Exponents

• Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power

• Example:

n

nn

b

a

b

a

23

x

2

23

x 2

9

x

Rules of Exponents

• Don’t Make Up Your Own Rules

• Many people try to make these rules:

• Proof:

nnn baba

222 2323

nnn baba

!!!!TRUE! NOT

!!!!TRUE! NOT

222 2323

Using Combinations of Rules to Simplify Expression with Exponents• Examples:

43225 pm 128425 pm 128165 pm 12880 pm

3325 yx 9635 yx 96125 yx

232332 32 yxyx 6496 98 yxyx 151072 yx

252

332

3

2

yx

yx

104

96

9

8

yx

yx

y

x

9

8 2

Homework Problems

• Section: 4.1

• Page: 261

• Problems: Odd: 5 – 11, 25 – 79

• MyMathLab Section 4.1 for practice

• MyMathLab Homework Quiz 4.1 is due for a grade on the date of our next class meeting

Integer Exponents

• Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, …

• An exponent of this type tells us how many times to multiply the base by itself

• Next we will learn the meaning of zero and negative integer exponents

• Examples: 0532

Integer Exponents

• Before giving the definition of zero and negative integer exponents, consider the pattern: 1624

823 422 221

02

12

22

1

2

1

4

1

8134 2733 932 331

03 13

23

1

3

1

9

1

1

2

1

2

2

1

1

3

1

2

3

1

Definition of Integer Exponents

• The patterns on the previous slide suggest the following definitions:

• These definitions work for any base, , that is not zero:

10 an

n

aa

1

a

05 1 32

3

2

1

8

1

Quotient Rule for Exponential Expressions

• When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent

Examples:

.

nmn

m

aa

a

7

4

5

5

4

12

x

x

374 55

8412 xx

“Slide Rule” for Exponential Expressions

• When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponentExample: Use rule to slide all factors to other part of the fraction:

• This rule applies to all types of exponents• Often used to make all exponents positive

sr

nm

dc

banm

sr

ba

dc

Simplify the Expression:(Show answer with positive exponents)

141

23

2

8

yy

yy

141

26

2

8

yy

yy

31

8

2

8

y

y

83

128

yy 11

16

y

Homework Problems

• Section: 4.2

• Page: 270

• Problems: Odd: 1 – 51, 57 – 77



Scientific Notation

• A number is written in scientific notation when it is in the form:

Examples:

• Note: When in scientific notation, a single non-zero digit precedes the decimal point

integeran is and 101 where,10 na a n

5102.3 9105342.1

201098.6

Converting from Normal Decimal Notation to Scientific Notation

• Given a decimal number:– Move the decimal to the right of the first non-zero digit

to get the “a”– Count the number of places the decimal was moved

• If it was moved to the right “n” places, use “-n” as the exponent on 10

• If it was moved to the left “n” places, use “n” as the exponent on 10

• Examples:

.

5102.3 9105342.1

201098.6

000,320 left places 5 decimal Move

3420000000015.0 right places 9 decimal Move

000,000,000,000,000,000,698 left places 20 decimal Move

na 10

Converting from Scientific Notation to Decimal Notation

• Given a number in scientific notation:– Move the decimal in “a” to the right “n” places,

if “n” is positive– Move the decimal in “a” to the left “n” places,

if “n” is negative

• Examples:

.

5102.3 9105342.1

201098.6

000,320right places 5 decimal Move

3420000000015.0left places 9 decimal Move

000,000,000,000,000,000,698right places 20 decimal Move

na 10

Applications of Scientific Notation

• Scientific notation is often used in situations where the numbers involved are extremely large or extremely small

• In doing calculations involving multiplication and/or division of numbers in scientific notation it is best to use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of 10 separately and to use rules of exponents to end up with an answer in scientific notation

• It is also common to round the answer to the least number of decimals seen in any individual number

Example of Calculations Involving Scientific Notation

• Perform the following calculations, round the answer to the appropriate number of places and in scientific notation

9

205

1053.1

1098.6102.3

9

205

10

1010

53.1

98.62.3

9205 10101059869281.14 341059869281.14

3510459869281.1 35105.1

notation? scientificin put this todo toneed wedoWhat

Homework Problems

• Section: 4.3

• Page: 278

• Problems: Odd: 1 – 9, 13 – 49, 63 – 75



Review of Terminology of Algebra

• Constant – A specific number

Examples of constants:

• Variable – A letter or other symbol used to represent a number whose value varies or is unknown

Examples of variables:

3 65

4

x n A


• Expression – constants and/or variables combined with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful wayExamples of expressions:

• Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

32 x5n

104 wy 92


• Term – an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variables

Examples of terms:

• Note: When constants and variables are multiplied, or when two variables are multiplied, it is common to omit the multiplication symbol

Previous example is commonly written:

2 m 25 x 23 yx 5

3

2yx

2 m 25x 23 yx 5

3

2xy


• Every term has a “coefficient”

• Coefficient – the constant factor of a term– (If no constant is seen, it is assumed to be 1)

• What is the coefficient of each of the following terms?

2

m

25x2

1

5

1 3

223 yx

5

3

2xy

Terminology of Algebra

• Every term has a “degree”

• Degree – the sum of the exponents on the variables in the term– (constant terms always have degree 0)

• What is the degree of each of the following terms?

2

m

25x0

1

2

56

23 yx

5

3

2xy

Review of Like Terms

• Recall that a term is a constant, a variable, or a product of a constant and variables

• Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients

• Example of Like Terms:

yxandyx 22 73

Review of Like Terms

• Given the term:

• Which of the following are like terms to this one?

324. xy

322 yx34xy

235 yx3

2

1xy

Adding and Subtracting Like Terms

• When “like terms” are added or subtracted, the result is a like term and its coefficient is the sum or difference of the coefficients of the other terms

• Examples:

xxx 72 x4

yxxxyyxx 2222 26194 xyyxx 6206 22

Polynomial

• Polynomial – a finite sum of terms

• Examples:

456 2 xx ?many terms How 3?first term of Degree

term?second oft Coefficien2

5-642 53 yxyx ?many terms How 2

term?second of Degree?first term oft Coefficien

103

Special Names for Certain Polynomials

Number of Terms

One term:

Two terms:

Three terms:

Special Name

monomial

binomial

trinomial456 2 xx

642 53 yxyx

yx29

Evaluating Polynomials

• To “evaluate” a polynomial is to replace variables with parentheses containing specific numbers and simplify

• Evaluate the polynomial for : 4,3 yx

224 13

224 xy

23424

984

Adding and Subtracting Polynomials

• To add or subtract polynomials horizontally:– Distribute to get rid of parentheses– Combine like terms

• Example:

233132 22 xxxxx

233132 22 xxxxx

xx 53 2

Adding and Subtracting Polynomials

• To add or subtract polynomials vertically:– Line up like terms in vertical columns– Add or subtract terms in each column

• Example: 23132 22 xxx

2 3

132 2

2

x

xx

332 xx

Homework Problems

• Section: 4.4• Page: 289• Problems: Odd: 1 – 55, 59 – 69,

73 – 77

• MyMathLab Section 4.4 for practice• MyMathLab Homework Quiz 4.4 is due for a

grade on the date of our next class meeting

Multiplying Polynomials

• To multiply polynomials: – Get rid of parentheses by multiplying every

term of the first by every term of the second using the rules of exponents

– Combine like terms

• Examples:

4523 2 xxx 12156452 223 xxxxx 12112 23 xxx

4532 xx 1215810 2 xxx 12710 2 xx

Multiplying Binomials by FOIL

• As seen by the last example, we already know how to multiply binomials by the general rule (every term of first by every term of the second)

• With binomials, this is sometimes called the FOIL method:– First times First– Outside times Outside– Inside times Inside– Last times Last

4532 xx 1215810 2 xxx 12710 2 xx

F O I L

Homework Problems

• Section: 4.5

• Page: 297

• Problems: Odd: 1 – 55, 61 – 83



Squaring a Binomial

• To square a binomial means to multiply it by itself

• Although a binomial can be squared by foiling it by itself, it is best to memorize a shortcut for squaring a binomial:

232x 3232 xx 9664 2 xxx 9124 2 xx

2ba 22 2 baba

232x

22 secondecond)2(first)(sfirst

9124 2 xx

Finding Higher Powers of Binomials

• To find powers of binomials higher than the second we use the definition of exponents and the rules already learned

• Example:

332x 3232 2 xx 329124 2 xxx

27183624128 223 xxxxx

2754368 23 xxx

Conjugate Binomials

• Two binomials are called “conjugates” if they are exactly the same except for the sign in the middle

• Examples: What is the conjugate of the given binomial?

:is Conjugate 32 x

:is Conjugate 45 x

32 x

45 x

Multiplying Conjugate Binomials

• Conjugate binomials can be multiplied by foil:

• However, it is best to memorize a formula for multiplying conjugate binomials:

3232 xx 9664 2 xxx 94 2 x

22 bababa lastlast and rst)(first)(fi

3232 xx 94 2 x

Homework Problems

• Section: 4.6

• Page: 303

• Problems: Odd: 3 – 19, 25 – 53



Dividing a Polynomial by a Monomial

• Write problem so that each term of the polynomial is individually placed over the monomial in “fraction form”

• Simplify each fraction by dividing out common factors xyxyxyyx 224128 23

xyxy

xy

xy

xy

xy

yx

2

2

2

4

2

12

2

8 23

xyyx

1264 2

Dividing a Polynomial by a Polynomial

• First write each polynomial in descending powers

• If a term of some power is missing, write that term with a zero coefficient

• Complete the problem exactly like a long division problem in basic math

Example 415032 232 xxx

40150023 223 xxxxx

150023 23 xxxx3

xxx 1203 23 402 xx

( )

xx 122 2 150

2

8 0 2 2 xx( )

15812 x

4

158122

x

x

Homework Problems

• Section: 4.7

• Page: 312

• Problems: Odd: 7 – 31, 39 – 75



elementary algebra exam 4 material exponential expressions & polynomials

Documents

slide rule

exponents positive slide

rules of exponents power

previous slide

positive exponents

power rule

class meeting slide

types of exponents