Chapter 6Chapter 6Review Review

PolynomialsPolynomials

2

Practice Product of Powers Property:

• Try:

• Try:

325 nnn

45 xx

3

Answers To Practice Problems

1. Answer:

2. Answer:

94545 xxxx

10325325 nnnnn

4

Practice Using the Power of a Power Property

1. Try:

2. Try:

44 )( p

54 )(n

5

Answers to Practice Problems

1. Answer:

2. Answer:

164444 )( ppp

205454 )( nnn

6

Practice Power of a Product Property

1. Try:

2. Try:

6)2( mn

4)(abc

7

Answers to Practice Problems

1. Answer:

2. Answer:

666666 642)2( nmnmmn

4444)( cbaabc

8

Practice Making Negative Exponents Positive

1. Try:

2. Try:

3d

5

1z

9

Answers to Negative Exponents Practice

1. Answer:

2. Answer:

33 1

dd

55

5 1

1z

z

z

10

Practice Rewriting the Expressions with Positive

Exponents:

1. Try:

2. Try:

zyx 3213

dcba 4324

11

Answers

1. Answer

2. Answer

32321

33

yx

zzyx

42

3432 4

4ca

dbdcba

12

Practice Quotient of Powers Property

1. Try:

2. Try:

3

9

a

a

4

3

y

y

13

Answers

1. Answer:

2. Answer:

639

3

9

1a

a

a

a

yyy

y 11344

3

Let’s Try Some

5

3

32

4

3816

y

5.3

4

3

1158

x

Hint: convert to a fraction rather than a decimal!

Answers are on the next slide!!!

Let’s Try Some

5

3

32

4

3816

y

5.3

4

3

1158

x

• Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.

• Polynomials – one or more monomials added or subtracted

• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.

Vocabulary

Like TermsLike Terms

Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.

Which terms are like? 3a2b, 4ab2, 3ab, -5ab2

4ab2 and -5ab2 are like.

Even though the others have the same variables, the exponents are not the same.

3a2b = 3aab, which is different from 4ab2 = 4abb.

Like TermsLike Terms

Constants are like terms.

Which terms are like? 2x, -3, 5b, 0

-3 and 0 are like.

Which terms are like? 3x, 2x2, 4, x

3x and x are like.

Which terms are like? 2wx, w, 3x, 4xw

2wx and 4xw are like.

A polynomial with only one term is called a monomial. A polynomial with two terms is

called a binomial. A polynomial with three terms is called a trinomial. Identify the

following polynomials:

Classifying Polynomials

Polynomial DegreeClassified by degree

Classified by number of terms

6

–2 x

3x + 1

–x 2 + 2 x – 5

4x 3 – 8x

2 x 4 – 7x

3 – 5x + 1

0

1

1

4

2

3

constant

linear

linear

quartic

quadratic

cubic

monomial

monomial

binomial

polynomial

trinomial

binomial

Add: (x2 + 3x + 1) + (4x2 +5)

Step 1: Underline like terms:

Step 2: Add the coefficients of like terms, do not change the powers of the variables:

Adding PolynomialsAdding Polynomials

(x2 + 3x + 1) + (4x2 +5)

Notice: ‘3x’ doesn’t have a like term.

(x2 + 4x2) + 3x + (1 + 5)

5x2 + 3x + 6

Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms!

Adding PolynomialsAdding Polynomials

(x2 + 3x + 1) + (4x2 +5)

5x2 + 3x + 6

(x2 + 3x + 1)

+ (4x2 +5)

Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)

(2a2+3ab+4b2) + (7a2+ab+-2b2)(2a2 + 3ab + 4b2)

+ (7a2 + ab + -2b2)

9a2 + 4ab + 2b2

Adding PolynomialsAdding Polynomials

• Add the following polynomials; you may stack them if you prefer:

Subtract: (3x2 + 2x + 7) - (x2 + x + 4)

Subtracting PolynomialsSubtracting Polynomials

Step 1: Change subtraction to addition (Keep-Change-Change.).

Step 2: Underline OR line up the like terms and add.

(3x2 + 2x + 7) + (- x2 + - x + - 4)

(3x2 + 2x + 7)

+ (- x2 + - x + - 4)

2x2 + x + 3

Subtracting PolynomialsSubtracting Polynomials

• Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)

Combine your like terms.

6y + x + 7a

Combine your like terms.

3a2 + 7ab + 5b2

2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2)

Combine your like terms. x2 - 3xy + 5y2

3. Add the following polynomials

(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)

Rewrite subtraction as adding the opposite.

(9y - 7x + 15a) + (+ 3y - 8x + 8a)

Combine the like terms.

9y + 3y - 7x - 8x + 15a + 8a

12y - 15x + 23a

4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

Rewrite subtraction as adding the opposite.

(7a - 10b) + (- 3a - 4b)Combine the like terms.

7a - 3a - 10b - 4b4a - 14b

5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)

Distribute your negative, and combine like terms

7x2 - xy + y2

6. Subtract the following polynomials

(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 9b

Find the sum. Write the answer in standard format.

(5x 3 – x + 2 x

2 + 7) + (3x 2 + 7 – 4 x) + (4x

2 – 8 – x 3)

Adding Polynomials

SOLUTION

Vertical format: Write each expression in standard form. Align like terms.

5x 3 + 2 x

2 – x + 7

3x 2 – 4 x + 7

– x 3 + 4x

2 – 8+

4x 3 + 9x

2 – 5x + 6

Find the sum. Write the answer in standard format.

(2 x 2 + x – 5) + (x + x

2 + 6)

Adding Polynomials

SOLUTION

Horizontal format: Add like terms.

(2 x 2 + x – 5) + (x + x

2 + 6) =(2 x 2 + x

2) + (x + x) + (–5 + 6)

=3x 2 + 2 x + 1

Find the difference.

(3x 2 – 5x + 3) – (2 x

2 – x – 4)

Subtracting Polynomials

SOLUTION

(3x 2 – 5x + 3) – (2 x

2 – x – 4)

= (3x 2 – 5x + 3) + (–1)(2 x

2 – x – 4)

= x 2 – 4x + 7

= (3x 2 – 5x + 3) – 2 x

2 + x + 4

= (3x 2 – 2 x

2) + (– 5x + x) + (3 + 4)

MultiplyingMultiplyingPolynomialsPolynomials

Distribute

Polynomials * Polynomials Polynomials * Polynomials

Multiplying a Polynomial by another Polynomial requires more than one distributing step.

Multiply: (2a + 7b)(3a + 5b)

Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):

6a2 + 10ab 21ab + 35b2

Then add those products, adding like terms:

6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2

Polynomials * Polynomials Polynomials * Polynomials

An alternative is to stack the polynomials and do long multiplication.

(2a + 7b)(3a + 5b)

6a2 + 10ab21ab + 35b2

(2a + 7b)

x (3a + 5b)

Multiply by 5b, then by 3a:(2a + 7b)

x (3a + 5b)When multiplying by 3a, line up the first term under 3a.

+

Add like terms: 6a2 + 31ab + 35b2

Polynomials * Polynomials Polynomials * Polynomials Multiply the following polynomials:

(x + 5)

x (2x + -1)

-x + -5

2x2 + 10x+

2x2 + 9x + -5

(3w + -2)

x (2w + -5)-15w + 10

6w2 + -4w+

6w2 + -19w + 10

Polynomials * Polynomials Polynomials * Polynomials

(2a2 + a + -1)

x (2a2 + 1)

2a2 + a + -1

4a4 + 2a3 + -2a2+

4a4 + 2a3 + a + -1

There is an acronym to help us remember how to multiply two binomials without stacking them.

Multiply please:Multiply please:

(2x + -3)(4x + 5)

(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15

The next slide has all the answers

Try multiplying these, Try multiplying these,

1) (3a + 4)(2a + 1)

2) (x + 4)(x - 5)

3) (x + 5)(x - 5)

4) (c - 3)(2c - 5)

5) (2w + 3)(2w - 3)

AnswersAnswers

1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4

2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20

3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25

4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15

5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9

1) Multiply. (2x + 3)(5x + 8)

Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8).

10x2 + 16x + 15x + 24

Combine like terms.

10x2 + 31x + 24

Multiply (y + 4)(y – 3)1. y2 + y – 12

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12

5. y2 + y + 12

6. y2 – y + 12

7. y2 + 7y + 12

8. y2 – 7y + 12

Multiply (2a – 3b)(2a + 4b)1. 4a2 + 14ab – 12b2

2. 4a2 – 14ab – 12b2

3. 4a2 + 8ab – 6ba – 12b2

4. 4a2 + 2ab – 12b2

5. 4a2 – 2ab – 12b2

5) Multiply (2x - 5)(x2 - 5x + 4)You must use the distributive property.

2x(x2 - 5x + 4) - 5(x2 - 5x + 4)

2x3 - 10x2 + 8x - 5x2 + 25x - 20

Group and combine like terms.

2x3 - 10x2 - 5x2 + 8x + 25x - 20

2x3 - 15x2 + 33x - 20

Multiply (2p + 1)(p2 – 3p + 4)1. 2p3 + 2p3 + p + 4

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12

Example: (x – 6)(2x + 1)

x(2x) + x(1) – (6)2x – 6(1)

2x2 + x – 12x – 6

2x2 – 11x – 6

2x2(3xy + 7x – 2y)

2x2(3xy) + 2x2(7x) + 2x2(–2y)

2x2(3xy + 7x – 2y)

6x3y + 14x2 – 4x2y

(x + 4)(x – 3)

(x + 4)(x – 3)

x(x) + x(–3) + 4(x) + 4(–3)

x2 – 3x + 4x – 12

x2 + x – 12

(2y – 3x)(y – 2)

(2y – 3x)(y – 2)

2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)

2y2 – 4y – 3xy + 6x

Multiply (2a + 3)2

1. 4a2 – 9

2. 4a2 + 9

3. 4a2 + 36a + 9

4. 4a2 + 12a + 9

Multiply (x – y)2

1. x2 + 2xy + y2

2. x2 – 2xy + y2

3. x2 + y2

4. x2 – y2

6) Multiply: (y – 2)(y + 2)(y)2 – (2)2

y2 – 4

7) Multiply: (5a + 6b)(5a – 6b)

(5a)2 – (6b)2

25a2 – 36b2

Multiply (4m – 3n)(4m + 3n)

1. 16m2 – 9n2

2. 16m2 + 9n2

3. 16m2 – 24mn - 9n2

4. 16m2 + 24mn + 9n2

Simplify.1)

2)

2(x 5)

2(m 2)

(x 5)(x 5) 2x 10x 25

(m 2)(m 2) 2m 4m 4

Difference of Squares.

Multiply.

1)

2)

3)

4)

(x 3)(x 3)

(m 7)(m 7)

(y 10)(y 10)

(t 8)(t 8)

2x 9 2m 49 2y 100

2t 64

Inner and Outer terms cancel!

Lesson Quiz: Part I

1. A square foot is 3–2 square yards. Simplify this

expression.

Simplify.

2. 2–6

3. (–7)–3

4. 60

5. –112

1

–121

Lesson Quiz: Part II

Evaluate each expression for the given value(s) of the variables(s).

6. x–4 for x =10

7. for a = 6 and b = 3

Lesson Quiz: Part I

Simplify each expression.

1.

2.

3.

4.

9

2

128

729

In an experiment, the approximate population P of a bacteria colony is given by

, where t is the number of days sincestart of the experiment. Find the population of the colony on the 8th day.

5.

480

Simplify. All variables represent nonnegative numbers.

6.

7.

Lesson Quiz: Part II

Lesson Quiz: Part I

Multiply.

1. (x + 7)2

2. (x – 2)2

3. (5x + 2y)2

4. (2x – 9y)2

5. (4x + 5y)(4x – 5y)

6. (m2 + 2n)(m2 – 2n)

x2 – 4x + 4

x2 + 14x + 49

25x2 + 20xy + 4y2

4x2 – 36xy + 81y2

16x2 – 25y2

m4 – 4n2

Lesson Quiz: Part II

7. Write a polynomial that represents the shaded area of the figure below.

14x – 85

x + 6

x – 6x – 7

x – 7