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1.(−3b2)(2b5)

2.(−6xy2)(−5x3y2)

3.(9x2yz5)(−4x3yz5)

4.(−ab2)(3ac3)

5.(5ab)(8a2)

6. −x2(3xy)(−6x3y2)

7. −6w3(2wx2)(3wx4)

8.(2x3)2

9.(5b2)(2b3)2

10.(−t3)(3rt2)3

Fill in the blanks with the missing monomial factor.

11.(2x3)(___) = −6x5

12.(−3b2)(___) = 12b7

13.(___)(3x2y) = −15x6y3

14.(___)(−2z4) = 6x2z5

15.(6xy2)(___) = −3x3y3

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Multiplying with the distributive property

To multiply a monomial times a larger polynomial, use the distributive property.

−2x3(4x5 − 6x4 + 5x3 − 7x2 + 8x−2)

=(−2x3 · 4x5) +(−2x3 · −6x4) +(−2x3 · 5x3) +(−2x3 · −7x2) +(−2x3 · 8x) +(−2x3 · −2)

Then follow the rules for multiplying monomials.

(−2x3 · 4x5) +(−2x3 · −6x4) +(−2x3 · 5x3) +(−2x3 · −7x2) +(−2x3 · 8x) +(−2x3 · −2)

= −8x8 + 12x7 −10x6 + 14x5 −16x4 + 4x3

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EXERCISE 6.6

Multiply the polynomials and give your answers in simplest form.

1. 5a(2a2 + 3a)

2. −2x2(x2 − 3x −2)

3. 2y2(11y2 − 3y + 5)

4. −3b3(2b2 − 3b + 4)

5. xy(3x2 + 5xy − 2y2)

6. 5x2y(5x2 − 7xy + y2)

7. 8x(x + 2 y − 3z)

8. −5ab(a2 − b3)

10. −3a4b3c2(−3a2b + 2bc − 7a5c4)

Fill in the blanks with the missing factor.

11.(____)(x + 1) = 3x + 3

12. a(____) = ab − 5a

13.(____)(2x − y) = 8x − 4y

14. 7x(____) = 7x + 49x2

15.(____)(2a + b) = 4a2b + 2ab2

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Multiplying binomials

Multiplication of two binomials is accomplished by repeated application of the distributive property, but there is a convenient shortcut, known by the acronym FOIL.

The distributive rule

To multiply two binomials by distributing, treat the first binomial as the multiplier and distribute it to both terms of the second binomial.

(x + 5)(x − 3) = x(x + 5)−3(x + 5)

That gives you two smaller distributive multiplications. Distribute x to both terms of x + 5, distribute −3 to both terms of x + 5, and combine like terms.

x(x + 5)−3(x + 5) = x2 + 5x − 3x − 15 = x2 + 2x − 15

The FOIL rule

FOIL stands for First, Outer, Inner, Last and is a reminder of the four multiplications that must be performed to successfully multiply two binomials.

First:(2x − 3)(5x + 7) =(2x)(5x)

Outer:(2x − 3)(5x + 7) =(2x)(5x) +(2x)(7)

Inner:(2x − 3)(5x + 7) =(2x)(5x) +(2x)(7) +( −3)(5x)

Last:(2x − 3)(5x + 7) =(2x)(5x) +(2x)(7) +(− 3)(5x) +( −3)(7)

You will often find that there are like terms that can be combined after the four multiplications are performed.

(2 x − 3)(5 x + 7) =(2x)(5x) +(2x)(7) +(−3)(5x) +(−3)(7)

= 10x2 + 14x − 15x − 21

= 10x2 − x − 21

When the binomials you’re multiplying are the sum and difference of the same two terms, like(x + 5) and(x − 5), the inner and outer terms will add to 0, leaving you with a difference of squares.

(x + 5)(x − 5) = x2 − 5x + 5x − 25

= x2 − 25

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EXERCISE 6.7

Use the FOIL rule to multiply the binomials. Give your answers in simplest form.

1.(x + 8)(x + 2)

2.(y − 4)(y − 9)

3.(t − 2)(t + 6)

4.(2x + 8)(x − 3)

5.(y − 9)(3y + 1)

6.(5x − 6)(3x + 4)

7.(6x − 1)(x + 5)

8.(1 − 3b)(5 + 2b)

9.(3x − 7)(2x + 5)

10.(5 − 2x)(5x − 2)

Try to predict the product of each pair of binomials without actually multiplying; then check with the FOIL rule.

11.(x − 4)(x + 4)

12.(x + 3)(x − 3)

13.(2x − 1)(2x + 1)

14.(3x + 5)(3x − 5)

15.(7 − 3x)(7 + 3x)

Fill in the blank with the missing term. Check your answer by FOILing.

16.(x + 3)(x + ____) = x2 + 5x + 6

17.(x − 7)(x − ____) = x2 − 9 x + 14

18.(2a + 1)(a + ____) = 2a2 + 9a + 4

19.(3x − 2)(x − ____) = 3x2 − 17x + 10

20.(2t + 3)(3t − ____) = 6t2 − t − 15

* * *

Multiplying larger polynomials

When one of the polynomials to be multiplied has more than two terms, it may be convenient to place them one under another, usually with the longer one on top and shorter one on the bottom, and multiply each term in the bottom polynomial by each term in the top polynomial, arranging like terms under one another in the result for easy combination. This is the same algorithm you learned for multiplying numbers with more than two digits.

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EXERCISE 6.8

Multiply