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# 5.E: Polynomial and Polynomial Functions (Exercises)

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5.1: Add and Subtract Polynomials
5.2: Properties of Exponents and Scientific Notation
5.3: Multiply Polynomials

## Practice Makes Perfect

Multiply Monomials

In the following exercises, multiply the monomials.

ⓐ (6y7)(−3y4)(6y7)(−3y4)
ⓑ (47rs2)(14rs3)(47rs2)(14rs3)

ⓐ (−10x5)(−3x3)(−10x5)(−3x3)
ⓑ(58x3y)(24x5y)(58x3y)(24x5y)

ⓐ30x830x8 ⓑ 15x8y215x8y2

ⓐ(−8u6)(−9u)(−8u6)(−9u)
ⓑ(23x2y)(34xy2)(23x2y)(34xy2)

ⓐ(−6c4)(−12c)(−6c4)(−12c)
ⓑ(35m3n2)(59m2n3)(35m3n2)(59m2n3)

ⓐ 72c572c5 ⓑ 13m5n513m5n5

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

ⓐ−8x(x2+2x−15)−8x(x2+2x−15)
ⓑ5pq3(p2−2pq+6q2)5pq3(p2−2pq+6q2)

ⓐ−5t(t2+3t−18);−5t(t2+3t−18);
ⓑ 9r3s(r2−3rs+5s2)9r3s(r2−3rs+5s2)

ⓐ −5t3−15t2+90t−5t3−15t2+90t
ⓑ 9sr5−27s2r4+45s3r39sr5−27s2r4+45s3r3

ⓐ−8y(y2+2y−15)−8y(y2+2y−15)
ⓑ −4y2z2(3y2+12yz−z2)−4y2z2(3y2+12yz−z2)

ⓐ−5m(m2+3m−18)−5m(m2+3m−18)
ⓑ−3x2y2(7x2+10xy−y2)−3x2y2(7x2+10xy−y2)

ⓐ −5m3−15m2+90m−5m3−15m2+90m
ⓑ −21x4y2−30x3y3+3x2y4−21x4y2−30x3y3+3x2y4

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

(w+5)(w+7)(w+5)(w+7)

(y+9)(y+3)(y+9)(y+3)

y2+12y+27y2+12y+27

(4p+11)(5p−4)(4p+11)(5p−4)

(7q+4)(3q−8)(7q+4)(3q−8)

21q2−44q−3221q2−44q−32

In the following exercises, multiply the binomials. Use any method.

(x+8)(x+3)(x+8)(x+3)

(y−6)(y−2)(y−6)(y−2)

y2−8y+12y2−8y+12

(2t−9)(10t+1)(2t−9)(10t+1)

(6p+5)(p+1)(6p+5)(p+1)

6p2+11p+56p2+11p+5

(q−5)(q+8)(q−5)(q+8)

(m+11)(m−4)(m+11)(m−4)

m2+7m−44m2+7m−44

(7m+1)(m−3)(7m+1)(m−3)

(3r−8)(11r+1)(3r−8)(11r+1)

33r2−85r−833r2−85r−8

(x2+3)(x+2)(x2+3)(x+2)

(y2−4)(y+3)(y2−4)(y+3)

y3+3y2−4y−12y3+3y2−4y−12

(5ab−1)(2ab+3)(5ab−1)(2ab+3)

(2xy+3)(3xy+2)(2xy+3)(3xy+2)

6x2y2+13xy+66x2y2+13xy+6

(x2+8)(x2−5)(x2+8)(x2−5)

(y2−7)(y2−4)(y2−7)(y2−4)

y4−11y2+28y4−11y2+28

(6pq−3)(4pq−5)(6pq−3)(4pq−5)

(3rs−7)(3rs−4)(3rs−7)(3rs−4)

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

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

(u+4)(u2+3u+2)(u+4)(u2+3u+2)

u3+7u2+14u+8u3+7u2+14u+8

(y+8)(4y2+y−7)(y+8)(4y2+y−7)

(a+10)(3a2+a−5)(a+10)(3a2+a−5)

3a3+31a2+5a−503a3+31a2+5a−50

(y2−3y+8)(4y2+y−7)(y2−3y+8)(4y2+y−7)

(2a2−5a+10)(3a2+a−5)(2a2−5a+10)(3a2+a−5)

6a4−13a3+15a2+35a−506a4−13a3+15a2+35a−50

Multiply Special Products

In the following exercises, multiply. Use either method.

(w−7)(w2−9w+10)(w−7)(w2−9w+10)

(p−4)(p2−6p+9)(p−4)(p2−6p+9)

p3−10p2+33p−36p3−10p2+33p−36

(3q+1)(q2−4q−5)(3q+1)(q2−4q−5)

(6r+1)(r2−7r−9)(6r+1)(r2−7r−9)

6r3−41r2−61r−96r3−41r2−61r−9

In the following exercises, square each binomial using the Binomial Squares Pattern.

(w+4)2(w+4)2

(q+12)2(q+12)2

q2+24q+144q2+24q+144

(3x−y)2(3x−y)2

(2y−3z)2(2y−3z)2

4y2−12yz+9z24y2−12yz+9z2

(y+14)2(y+14)2

(x+23)2(x+23)2

x2+43x+49x2+43x+49

(15x−17y)2(15x−17y)2

(18x−19y)2(18x−19y)2

164x2−136xy+181y2164x2−136xy+181y2

(3x2+2)2(3x2+2)2

(5u2+9)2(5u2+9)2

25u4+90u2+8125u4+90u2+81

(4y3−2)2(4y3−2)2

(8p3−3)2(8p3−3)2

64p6−48p3+964p6−48p3+9

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

(5k+6)(5k−6)(5k+6)(5k−6)

(8j+4)(8j−4)(8j+4)(8j−4)

64j2−1664j2−16

(11k+4)(11k−4)(11k+4)(11k−4)

(9c+5)(9c−5)(9c+5)(9c−5)

81c2−2581c2−25

(9c−2d)(9c+2d)(9c−2d)(9c+2d)

(7w+10x)(7w−10x)(7w+10x)(7w−10x)

49w2−100x249w2−100x2

(m+23n)(m−23n)(m+23n)(m−23n)

(p+45q)(p−45q)(p+45q)(p−45q)

p2−1625q2p2−1625q2

(ab−4)(ab+4)(ab−4)(ab+4)

(xy−9)(xy+9)(xy−9)(xy+9)

(12p3−11q2)(12p3+11q2)(12p3−11q2)(12p3+11q2)

(15m2−8n4)(15m2+8n4)(15m2−8n4)(15m2+8n4)

225m4−64n8225m4−64n8

In the following exercises, find each product.

(p−3)(p+3)(p−3)(p+3)

(t−9)2(t−9)2

t2−18t+81t2−18t+81

(m+n)2(m+n)2

(2x+y)(x−2y)(2x+y)(x−2y)

2x2−3xy−2y22x2−3xy−2y2

(2r+12)2(2r+12)2

(3p+8)(3p−8)(3p+8)(3p−8)

9p2−649p2−64

(7a+b)(a−7b)(7a+b)(a−7b)

(k−6)2(k−6)2

k2−12k+36k2−12k+36

(a5−7b)2(a5−7b)2

(x2+8y)(8x−y2)(x2+8y)(8x−y2)

8x3−x2y2+64xy−8y38x3−x2y2+64xy−8y3

(r6+s6)(r6−s6)(r6+s6)(r6−s6)

(y4+2z)2(y4+2z)2

y8+4y4z+4z2y8+4y4z+4z2

(x5+y5)(x5−y5)(x5+y5)(x5−y5)

(m3−8n)2(m3−8n)2

m6−16m3n+64n2m6−16m3n+64n2

(9p+8q)2(9p+8q)2

(r2−s3)(r3+s2)(r2−s3)(r3+s2)

r5+r2s2−r3s3−s5r5+r2s2−r3s3−s5

## Mixed Practice

(10y−6)+(4y−7)(10y−6)+(4y−7)

(15p−4)+(3p−5)(15p−4)+(3p−5)

18p−918p−9

(x2−4x−34)−(x2+7x−6)(x2−4x−34)−(x2+7x−6)

(j2−8j−27)−(j2+2j−12)(j2−8j−27)−(j2+2j−12)

−10j−15−10j−15

(15f8)(20f3)(15f8)(20f3)

(14d5)(36d2)(14d5)(36d2)

9d79d7

(4a3b)(9a2b6)(4a3b)(9a2b6)

(6m4n3)(7mn5)(6m4n3)(7mn5)

42m5n842m5n8

−5m(m2+3m−18)−5m(m2+3m−18)

5q3(q2−2q+6)5q3(q2−2q+6)

5q5−10q4+30q35q5−10q4+30q3

(s−7)(s+9)(s−7)(s+9)

(y2−2y)(y+1)(y2−2y)(y+1)

y3−y2−2yy3−y2−2y

(5x−y)(x−4)(5x−y)(x−4)

(6k−1)(k2+2k−4)(6k−1)(k2+2k−4)

6k3+11k2−26k+46k3+11k2−26k+4

(3x−11y)(3x−11y)(3x−11y)(3x−11y)

(11−b)(11+b)(11−b)(11+b)

121−b2121−b2

(rs−27)(rs+27)(rs−27)(rs+27)

(2x2−3y4)(2x2+3y4)(2x2−3y4)(2x2+3y4)

4x4−9y84x4−9y8

(m−15)2(m−15)2

(3d+1)2(3d+1)2

9d2+6d+19d2+6d+1

(4a+10)2(4a+10)2

(3z+15)2(3z+15)2

9z2+65z+1259z2+65z+125

Multiply Polynomial Functions

For functions f(x)=x+2f(x)=x+2 and g(x)=3x2−2x+4,g(x)=3x2−2x+4, find ⓐ (f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(−1)(f·g)(−1)

For functions f(x)=x−1f(x)=x−1 and g(x)=4x2+3x−5,g(x)=4x2+3x−5, find ⓐ (f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(−2)(f·g)(−2)

ⓐ (f⋅g)(x)=4x3−x2−8x+5(f·g)(x)=4x3−x2−8x+5
ⓑ (f⋅g)(−2)=−15(f·g)(−2)=−15

For functions f(x)=2x−7f(x)=2x−7 and g(x)=2x+7,g(x)=2x+7, find ⓐ(f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(−3)(f·g)(−3)

For functions f(x)=7x−8f(x)=7x−8 and g(x)=7x+8,g(x)=7x+8, find ⓐ (f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(−2)(f·g)(−2)

ⓐ (f⋅g)(x)=49x2−64(f·g)(x)=49x2−64
ⓑ (f⋅g)(−2)=187(f·g)(−2)=187

For functions f(x)=x2−5x+2f(x)=x2−5x+2 and g(x)=x2−3x−1,g(x)=x2−3x−1, find ⓐ (f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(−1)(f·g)(−1)

For functions f(x)=x2+4x−3f(x)=x2+4x−3 and g(x)=x2+2x+4,g(x)=x2+2x+4, find ⓐ (f⋅g)(x)(f·g)(x) ⓑ (f⋅g)(1)(f·g)(1)

ⓐ (f⋅g)(x)=x4+6x3+9x2+10x−12(f·g)(x)=x4+6x3+9x2+10x−12 ⓑ (f⋅g)(1)=14(f·g)(1)=14

## Writing Exercises

Which method do you prefer to use when multiplying two binomials: the Distributive Property or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the Distributive Property or the Vertical Method? Why?

Multiply the following:

(x+2)(x−2)(y+7)(y−7)(w+5)(w−5)(x+2)(x−2)(y+7)(y−7)(w+5)(w−5)

Explain the pattern that you see in your answers.

Multiply the following:

(p+3)(p+3)(q+6)(q+6)(r+1)(r+1)(p+3)(p+3)(q+6)(q+6)(r+1)(r+1)

Explain the pattern that you see in your answers.

Why does (a+b)2(a+b)2 result in a trinomial, but (a−b)(a+b)(a−b)(a+b) result in a binomial?