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Mathematics LibreTexts

5.E: Polynomial and Polynomial Functions (Exercises)

  • Page ID
    17629
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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.

    Answers will vary.

    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?

    Answers will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    .

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

    5.4: Dividing Polynomials