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Mathematics LibreTexts

6.3E: Exercises

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    30425
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    Practice Makes Perfect

    Multiply a Polynomial by a Monomial

    In the following exercises, multiply.

    Exercise \(\PageIndex{1}\)

    4\((w+10)\)

    Answer

    4w+40

    Exercise \(\PageIndex{2}\)

    6(b+8)

    Exercise \(\PageIndex{3}\)

    −3(a+7)

    Answer

    −3a−21

    Exercise \(\PageIndex{4}\)

    −5(p+9)

    Exercise \(\PageIndex{5}\)

    2(x−7)

    Answer

    2x−14

    Exercise \(\PageIndex{6}\)

    7(y−4)

    Exercise \(\PageIndex{7}\)

    −3(k−4)

    Answer

    −3k+12

    Exercise \(\PageIndex{8}\)

    −8(j−5)

    Exercise \(\PageIndex{9}\)

    q(q+5)

    Answer

    \(q^{2}+5 q\)

    Exercise \(\PageIndex{10}\)

    k(k+7)

    Exercise \(\PageIndex{11}\)

    −b(b+9)

    Answer

    \(-b^{2}-9 b\)

    Exercise \(\PageIndex{12}\)

    −y(y+3)

    Exercise \(\PageIndex{13}\)

    −x(x−10)

    Answer

    \(-x^{2}+10 x\)

    Exercise \(\PageIndex{14}\)

    −p(p−15)

    Exercise \(\PageIndex{15}\)

    6r(4r+s)

    Answer

    \(24 r^{2}+6 r s\)

    Exercise \(\PageIndex{16}\)

    5c(9c+d)

    Exercise \(\PageIndex{17}\)

    12x(x−10)

    Answer

    \(12 x^{2}-120 x\)

    Exercise \(\PageIndex{18}\)

    9m(m−11)

    Exercise \(\PageIndex{19}\)

    −9a(3a+5)

    Answer

    \(-27 a^{2}-45 a\)

    Exercise \(\PageIndex{20}\)

    −4p(2p+7)

    Exercise \(\PageIndex{21}\)

    3\(\left(p^{2}+10 p+25\right)\)

    Answer

    \(3 p^{2}+30 p+75\)

    Exercise \(\PageIndex{22}\)

    6\(\left(y^{2}+8 y+16\right)\)

    Exercise \(\PageIndex{23}\)

    \(-8 x\left(x^{2}+2 x-15\right)\)

    Answer

    \(-8 x^{3}-16 x^{2}+120 x\)

    Exercise \(\PageIndex{24}\)

    \(-5 t\left(t^{2}+3 t-18\right)\)

    Exercise \(\PageIndex{25}\)

    5\(q^{3}\left(q^{3}-2 q+6\right)\)

    Answer

    \(5 q^{6}-10 q^{4}+30 q^{3}\)

    Exercise \(\PageIndex{26}\)

    4\(x^{3}\left(x^{4}-3 x+7\right)\)

    Exercise \(\PageIndex{27}\)

    \(-8 y\left(y^{2}+2 y-15\right)\)

    Answer

    \(-8 y^{3}-16 y^{2}+120 y\)

    Exercise \(\PageIndex{28}\)

    \(-5 m\left(m^{2}+3 m-18\right)\)

    Exercise \(\PageIndex{29}\)

    5\(q^{3}\left(q^{2}-2 q+6\right)\)

    Answer

    \(5 q^{5}-10 q^{4}+30 q^{3}\)

    Exercise \(\PageIndex{30}\)

    9\(r^{3}\left(r^{2}-3 r+5\right)\)

    Exercise \(\PageIndex{31}\)

    \(-4 z^{2}\left(3 z^{2}+12 z-1\right)\)

    Answer

    \(-12 z^{4}-48 z^{3}+4 z^{2}\)

    Exercise \(\PageIndex{32}\)

    \(-3 x^{2}\left(7 x^{2}+10 x-1\right)\)

    Exercise \(\PageIndex{33}\)

    \((2 m-9) m\)

    Answer

    \(2 m^{2}-9 m\)

    Exercise \(\PageIndex{34}\)

    \((8 j-1) j\)

    Exercise \(\PageIndex{35}\)

    \((w-6) \cdot 8\)

    Answer

    \(8 w-48\)

    Exercise \(\PageIndex{36}\)

    \((k-4) \cdot 5\)

    Exercise \(\PageIndex{37}\)

    4\((x+10)\)

    Answer

    4x+40

    Exercise \(\PageIndex{38}\)

    6(y+8)

    Exercise \(\PageIndex{39}\)

    15(r−24)

    Answer

    15r−360

    Exercise \(\PageIndex{40}\)

    12(v−30)

    Exercise \(\PageIndex{41}\)

    −3(m+11)

    Answer

    −3m−33

    Exercise \(\PageIndex{42}\)

    −4(p+15)

    Exercise \(\PageIndex{43}\)

    −8(z−5)

    Answer

    −8z+40

    Exercise \(\PageIndex{44}\)

    −3(x−9)

    Exercise \(\PageIndex{45}\)

    u(u+5)

    Answer

    \(u^{2}+5 u\)

    Exercise \(\PageIndex{46}\)

    \(q(q+7)\)

    Exercise \(\PageIndex{47}\)

    \(n\left(n^{2}-3 n\right)\)

    Answer

    \(n^{3}-3 n^{2}\)

    Exercise \(\PageIndex{48}\)

    \(s\left(s^{2}-6 s\right)\)

    Exercise \(\PageIndex{49}\)

    6\(x(4 x+y)\)

    Answer

    \(24 x^{2}+6 x y\)

    Exercise \(\PageIndex{50}\)

    5a(9a+b)

    Exercise \(\PageIndex{51}\)

    5p(11p−5q)

    Answer

    \(55 p^{2}-25 p q\)

    Exercise \(\PageIndex{52}\)

    12\(u(3 u-4 v)\)

    Exercise \(\PageIndex{53}\)

    3\(\left(v^{2}+10 v+25\right)\)

    Answer

    \(3 v^{2}+30 v+75\)

    Exercise \(\PageIndex{54}\)

    6\(\left(x^{2}+8 x+16\right)\)

    Exercise \(\PageIndex{55}\)

    2\(n\left(4 n^{2}-4 n+1\right)\)

    Answer

    \(8 n^{3}-8 n^{2}+2 n\)

    Exercise \(\PageIndex{56}\)

    3\(r\left(2 r^{2}-6 r+2\right)\)

    Exercise \(\PageIndex{57}\)

    \(-8 y\left(y^{2}+2 y-15\right)\)

    Answer

    \(-8 y^{3}-16 y^{2}+120 y\)

    Exercise \(\PageIndex{58}\)

    \(-5 m\left(m^{2}+3 m-18\right)\)

    Exercise \(\PageIndex{59}\)

    5\(q^{3}\left(q^{2}-2 q+6\right)\)

    Answer

    \(5 q^{5}-10 q^{4}+30 q^{3}\)

    Exercise \(\PageIndex{60}\)

    9\(r^{3}\left(r^{2}-3 r+5\right)\)

    Exercise \(\PageIndex{61}\)

    \(-4 z^{2}\left(3 z^{2}+12 z-1\right)\)

    Answer

    \(-12 z^{4}-48 z^{3}+4 z^{2}\)

    Exercise \(\PageIndex{62}\)

    \(-3 x^{2}\left(7 x^{2}+10 x-1\right)\)

    Exercise \(\PageIndex{63}\)

    \((2 y-9) y\)

    Answer

    \(18 y^{2}-9 y\)

    Exercise \(\PageIndex{64}\)

    \((8 b-1) b\)

    Multiply a Binomial by a Binomial

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

    Exercise \(\PageIndex{65}\)

    (w+5)(w+7)

    Answer

    \(w^{2}+12 w+35\)

    Exercise \(\PageIndex{66}\)

    (y+9)(y+3)

    Exercise \(\PageIndex{67}\)

    (p+11)(p−4)

    Answer

    \(p^{2}+7 p-44\)

    Exercise \(\PageIndex{68}\)

    (q+4)(q−8)

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

    Exercise \(\PageIndex{69}\)

    (x+8)(x+3)

    Answer

    \(x^{2}+11 x+24\)

    Exercise \(\PageIndex{70}\)

    (y+7)(y+4)

    Exercise \(\PageIndex{71}\)

    (y−6)(y−2)

    Answer

    \(y^{2}-8 y+12\)

    Exercise \(\PageIndex{72}\)

    (x−7)(x−2)

    Exercise \(\PageIndex{73}\)

    (w−4)(w+7)

    Answer

    \(w^{2}+3 w-28\)

    Exercise \(\PageIndex{74}\)

    \((q-5)(q+8)\)

    Exercise \(\PageIndex{75}\)

    (p+12)(p−5)

    Answer

    \(p^{2}+7 p-60\)

    Exercise \(\PageIndex{76}\)

    (m+11)(m−4)

    Exercise \(\PageIndex{77}\)

    (6p+5)(p+1)

    Answer

    \(6 p^{2}+11 p+5\)

    Exercise \(\PageIndex{78}\)

    \((7 m+1)(m+3)\)

    Exercise \(\PageIndex{79}\)

    (2t−9)(10t+1)

    Answer

    \(20 t^{2}-88 t-9\)

    Exercise \(\PageIndex{80}\)

    (3r−8)(11r+1)

    Exercise \(\PageIndex{81}\)

    (5x−y)(3x−6)

    Answer

    \(15 x^{2}-3 x y-30 x+6 y\)

    Exercise \(\PageIndex{82}\)

    (10a−b)(3a−4)

    Exercise \(\PageIndex{83}\)

    (a+b)(2a+3b)

    Answer

    \(2 a^{2}+5 a b+3 b^{2}\)

    Exercise \(\PageIndex{84}\)

    (r+s)(3r+2s)

    Exercise \(\PageIndex{85}\)

    (4z−y)(z−6)

    Answer

    \(4 z^{2}-24 z-z y+6 y\)

    Exercise \(\PageIndex{86}\)

    (5x−y)(x−4)

    Exercise \(\PageIndex{87}\)

    \(\left(x^{2}+3\right)(x+2)\)

    Answer

    \(x^{3}+2 x^{2}+3 x+6\)

    Exercise \(\PageIndex{88}\)

    \(\left(y^{2}-4\right)(y+3)\)

    Exercise \(\PageIndex{89}\)

    \(\left(x^{2}+8\right)\left(x^{2}-5\right)\)

    Answer

    \(x^{4}+3 x^{2}-40\)

    Exercise \(\PageIndex{90}\)

    \(\left(y^{2}-7\right)\left(y^{2}-4\right)\)

    Exercise \(\PageIndex{91}\)

    (5ab−1)(2ab+3)

    Answer

    \(10 a^{2} b^{2}+13 a b-3\)

    Exercise \(\PageIndex{92}\)

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

    Exercise \(\PageIndex{93}\)

    (6pq−3)(4pq−5)

    Answer

    \(24 p^{2} q^{2}-42 p q+15\)

    Exercise \(\PageIndex{94}\)

    (3rs−7)(3rs−4)

    Multiply a Trinomial by a Binomial

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

    Exercise \(\PageIndex{95}\)

    \((x+5)\left(x^{2}+4 x+3\right)\)

    Answer

    \(x^{3}+9 x^{2}+23 x+15\)

    Exercise \(\PageIndex{96}\)

    \((u+4)\left(u^{2}+3 u+2\right)\)

    Exercise \(\PageIndex{97}\)

    \((y+8)\left(4 y^{2}+y-7\right)\)

    Answer

    \(4 y^{3}+33 y^{2}+y-56\)

    Exercise \(\PageIndex{98}\)

    \((a+10)\left(3 a^{2}+a-5\right)\)

    In the following exercises, multiply. Use either method.

    Exercise \(\PageIndex{99}\)

    \((w-7)\left(w^{2}-9 w+10\right)\)

    Answer

    \(w^{3}-16 w^{2}+73 w-70\)

    Exercise \(\PageIndex{100}\)

    \((p-4)\left(p^{2}-6 p+9\right)\)

    Exercise \(\PageIndex{101}\)

    \((3 q+1)\left(q^{2}-4 q-5\right)\)

    Answer

    \(3 q^{3}-11 q^{2}-19 q-5\)

    Exercise \(\PageIndex{102}\)

    \((6 r+1)\left(r^{2}-7 r-9\right)\)

    Mixed Practice

    Exercise \(\PageIndex{103}\)

    (10y−6)+(4y−7)

    Answer

    14y−13

    Exercise \(\PageIndex{104}\)

    (15p−4)+(3p−5)

    Exercise \(\PageIndex{105}\)

    \(\left(x^{2}-4 x-34\right)-\left(x^{2}+7 x-6\right)\)

    Answer

    −11x−28

    Exercise \(\PageIndex{106}\)

    \(\left(j^{2}-8 j-27\right)-\left(j^{2}+2 j-12\right)\)

    Exercise \(\PageIndex{107}\)

    5\(q\left(3 q^{2}-6 q+11\right)\)

    Answer

    \(15 q^{3}-30 q^{2}+55 q\)

    Exercise \(\PageIndex{108}\)

    8\(t\left(2 t^{2}-5 t+6\right)\)

    Exercise \(\PageIndex{109}\)

    (s−7)(s+9)

    Answer

    \(s^{2}+2 s-63\)

    Exercise \(\PageIndex{110}\)

    (x−5)(x+13)

    Exercise \(\PageIndex{111}\)

    \(\left(y^{2}-2 y\right)(y+1)\)

    Answer

    \(y^{3}-y^{2}-2 y\)

    Exercise \(\PageIndex{112}\)

    \(\left(a^{2}-3 a\right)(4 a+5)\)

    Exercise \(\PageIndex{113}\)

    \((3 n-4)\left(n^{2}+n-7\right)\)

    Answer

    \(3 n^{3}-n^{2}-25 n+28\)

    Exercise \(\PageIndex{114}\)

    \((6 k-1)\left(k^{2}+2 k-4\right)\)

    Exercise \(\PageIndex{115}\)

    \((7 p+10)(7 p-10)\)

    Answer

    \(49 p^{2}-100\)

    Exercise \(\PageIndex{116}\)

    (3y+8)(3y−8)

    Exercise \(\PageIndex{117}\)

    \(\left(4 m^{2}-3 m-7\right) m^{2}\)

    Answer

    \(4 m^{4}-3 m^{3}-7 m^{2}\)

    Exercise \(\PageIndex{118}\)

    \(\left(15 c^{2}-4 c+5\right) c^{4}\)

    Exercise \(\PageIndex{119}\)

    \((5 a+7 b)(5 a+7 b)\)

    Answer

    \(25 a^{2}+70 a b+49 b^{2}\)

    Exercise \(\PageIndex{120}\)

    (3x−11y)(3x−11y)

    Exercise \(\PageIndex{121}\)

    (4y+12z)(4y−12z)

    Answer

    \(16 y^{2}-144 z^{2}\)

    Everyday Math

    Exercise \(\PageIndex{122}\)

    Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as 10+3 and 15 as 10+5.

    1. Multiply (10+3)(10+5) by the FOIL method.
    2. Multiply 13·15 without using a calculator.
    3. Which way is easier for you? Why?

    Exercise \(\PageIndex{123}\)

    Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as 20−2 and 17 as 20−3.

    1. Multiply (20−2)(20−3) by the FOIL method.
    2. Multiply 18·17 without using a calculator.
    3. Which way is easier for you? Why?
    Answer
    1. 306
    2. 306
    3. Answers will vary.

    Writing Exercises

    Exercise \(\PageIndex{124}\)

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

    Exercise \(\PageIndex{125}\)

    Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?

    Answer

    Answers will vary.

    Exercise \(\PageIndex{126}\)

    Multiply the following:

    \(\begin{array}{l}{(x+2)(x-2)} \\ {(y+7)(y-7)} \\ {(w+5)(w-5)}\end{array}\)

    Explain the pattern that you see in your answers.

    Exercise \(\PageIndex{127}\)

    Multiply the following:

    \(\begin{array}{l}{(m-3)(m+3)} \\ {(n-10)(n+10)} \\ {(p-8)(p+8)}\end{array}\)

    Explain the pattern that you see in your answers.

    Answer

    Answers may vary.

    Exercise \(\PageIndex{128}\)

    Multiply the following:

    \(\begin{array}{l}{(p+3)(p+3)} \\ {(q+6)(q+6)} \\ {(r+1)(r+1)}\end{array}\)

    Explain the pattern that you see in your answers.

    Exercise \(\PageIndex{129}\)

    Multiply the following:

    \(\begin{array}{l}{(x-4)(x-4)} \\ {(y-1)(y-1)} \\ {(z-7)(z-7)}\end{array}\)

    Explain the pattern that you see in your answers.

    Answer

    Answers may vary.

    Self Check

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

    This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “multiply a polynomial by a monomial,” “multiply a binomial by a binomial,” and “multiply a trinomial by a binomial.” The rest of the cells are blank.

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