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# 6.3E: Exercises

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## Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Exercise $$\PageIndex{1}$$

4$$(w+10)$$

4w+40

Exercise $$\PageIndex{2}$$

6(b+8)

Exercise $$\PageIndex{3}$$

−3(a+7)

−3a−21

Exercise $$\PageIndex{4}$$

−5(p+9)

Exercise $$\PageIndex{5}$$

2(x−7)

2x−14

Exercise $$\PageIndex{6}$$

7(y−4)

Exercise $$\PageIndex{7}$$

−3(k−4)

−3k+12

Exercise $$\PageIndex{8}$$

−8(j−5)

Exercise $$\PageIndex{9}$$

q(q+5)

$$q^{2}+5 q$$

Exercise $$\PageIndex{10}$$

k(k+7)

Exercise $$\PageIndex{11}$$

−b(b+9)

$$-b^{2}-9 b$$

Exercise $$\PageIndex{12}$$

−y(y+3)

Exercise $$\PageIndex{13}$$

−x(x−10)

$$-x^{2}+10 x$$

Exercise $$\PageIndex{14}$$

−p(p−15)

Exercise $$\PageIndex{15}$$

6r(4r+s)

$$24 r^{2}+6 r s$$

Exercise $$\PageIndex{16}$$

5c(9c+d)

Exercise $$\PageIndex{17}$$

12x(x−10)

$$12 x^{2}-120 x$$

Exercise $$\PageIndex{18}$$

9m(m−11)

Exercise $$\PageIndex{19}$$

−9a(3a+5)

$$-27 a^{2}-45 a$$

Exercise $$\PageIndex{20}$$

−4p(2p+7)

Exercise $$\PageIndex{21}$$

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

$$3 p^{2}+30 p+75$$

Exercise $$\PageIndex{22}$$

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

Exercise $$\PageIndex{23}$$

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

$$-8 x^{3}-16 x^{2}+120 x$$

Exercise $$\PageIndex{24}$$

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

Exercise $$\PageIndex{25}$$

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

$$5 q^{6}-10 q^{4}+30 q^{3}$$

Exercise $$\PageIndex{26}$$

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

Exercise $$\PageIndex{27}$$

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

$$-8 y^{3}-16 y^{2}+120 y$$

Exercise $$\PageIndex{28}$$

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

Exercise $$\PageIndex{29}$$

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

$$5 q^{5}-10 q^{4}+30 q^{3}$$

Exercise $$\PageIndex{30}$$

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

Exercise $$\PageIndex{31}$$

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

$$-12 z^{4}-48 z^{3}+4 z^{2}$$

Exercise $$\PageIndex{32}$$

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

Exercise $$\PageIndex{33}$$

$$(2 m-9) m$$

$$2 m^{2}-9 m$$

Exercise $$\PageIndex{34}$$

$$(8 j-1) j$$

Exercise $$\PageIndex{35}$$

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

$$8 w-48$$

Exercise $$\PageIndex{36}$$

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

Exercise $$\PageIndex{37}$$

4$$(x+10)$$

4x+40

Exercise $$\PageIndex{38}$$

6(y+8)

Exercise $$\PageIndex{39}$$

15(r−24)

15r−360

Exercise $$\PageIndex{40}$$

12(v−30)

Exercise $$\PageIndex{41}$$

−3(m+11)

−3m−33

Exercise $$\PageIndex{42}$$

−4(p+15)

Exercise $$\PageIndex{43}$$

−8(z−5)

−8z+40

Exercise $$\PageIndex{44}$$

−3(x−9)

Exercise $$\PageIndex{45}$$

u(u+5)

$$u^{2}+5 u$$

Exercise $$\PageIndex{46}$$

$$q(q+7)$$

Exercise $$\PageIndex{47}$$

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

$$n^{3}-3 n^{2}$$

Exercise $$\PageIndex{48}$$

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

Exercise $$\PageIndex{49}$$

6$$x(4 x+y)$$

$$24 x^{2}+6 x y$$

Exercise $$\PageIndex{50}$$

5a(9a+b)

Exercise $$\PageIndex{51}$$

5p(11p−5q)

$$55 p^{2}-25 p q$$

Exercise $$\PageIndex{52}$$

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

Exercise $$\PageIndex{53}$$

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

$$3 v^{2}+30 v+75$$

Exercise $$\PageIndex{54}$$

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

Exercise $$\PageIndex{55}$$

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

$$8 n^{3}-8 n^{2}+2 n$$

Exercise $$\PageIndex{56}$$

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

Exercise $$\PageIndex{57}$$

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

$$-8 y^{3}-16 y^{2}+120 y$$

Exercise $$\PageIndex{58}$$

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

Exercise $$\PageIndex{59}$$

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

$$5 q^{5}-10 q^{4}+30 q^{3}$$

Exercise $$\PageIndex{60}$$

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

Exercise $$\PageIndex{61}$$

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

$$-12 z^{4}-48 z^{3}+4 z^{2}$$

Exercise $$\PageIndex{62}$$

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

Exercise $$\PageIndex{63}$$

$$(2 y-9) y$$

$$18 y^{2}-9 y$$

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)

$$w^{2}+12 w+35$$

Exercise $$\PageIndex{66}$$

(y+9)(y+3)

Exercise $$\PageIndex{67}$$

(p+11)(p−4)

$$p^{2}+7 p-44$$

Exercise $$\PageIndex{68}$$

(q+4)(q−8)

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

Exercise $$\PageIndex{69}$$

(x+8)(x+3)

$$x^{2}+11 x+24$$

Exercise $$\PageIndex{70}$$

(y+7)(y+4)

Exercise $$\PageIndex{71}$$

(y−6)(y−2)

$$y^{2}-8 y+12$$

Exercise $$\PageIndex{72}$$

(x−7)(x−2)

Exercise $$\PageIndex{73}$$

(w−4)(w+7)

$$w^{2}+3 w-28$$

Exercise $$\PageIndex{74}$$

$$(q-5)(q+8)$$

Exercise $$\PageIndex{75}$$

(p+12)(p−5)

$$p^{2}+7 p-60$$

Exercise $$\PageIndex{76}$$

(m+11)(m−4)

Exercise $$\PageIndex{77}$$

(6p+5)(p+1)

$$6 p^{2}+11 p+5$$

Exercise $$\PageIndex{78}$$

$$(7 m+1)(m+3)$$

Exercise $$\PageIndex{79}$$

(2t−9)(10t+1)

$$20 t^{2}-88 t-9$$

Exercise $$\PageIndex{80}$$

(3r−8)(11r+1)

Exercise $$\PageIndex{81}$$

(5x−y)(3x−6)

$$15 x^{2}-3 x y-30 x+6 y$$

Exercise $$\PageIndex{82}$$

(10a−b)(3a−4)

Exercise $$\PageIndex{83}$$

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

$$2 a^{2}+5 a b+3 b^{2}$$

Exercise $$\PageIndex{84}$$

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

Exercise $$\PageIndex{85}$$

(4z−y)(z−6)

$$4 z^{2}-24 z-z y+6 y$$

Exercise $$\PageIndex{86}$$

(5x−y)(x−4)

Exercise $$\PageIndex{87}$$

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

$$x^{3}+2 x^{2}+3 x+6$$

Exercise $$\PageIndex{88}$$

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

Exercise $$\PageIndex{89}$$

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

$$x^{4}+3 x^{2}-40$$

Exercise $$\PageIndex{90}$$

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

Exercise $$\PageIndex{91}$$

(5ab−1)(2ab+3)

$$10 a^{2} b^{2}+13 a b-3$$

Exercise $$\PageIndex{92}$$

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

Exercise $$\PageIndex{93}$$

(6pq−3)(4pq−5)

$$24 p^{2} q^{2}-42 p q+15$$

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 $$\PageIndex{95}$$

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

$$x^{3}+9 x^{2}+23 x+15$$

Exercise $$\PageIndex{96}$$

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

Exercise $$\PageIndex{97}$$

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

$$4 y^{3}+33 y^{2}+y-56$$

Exercise $$\PageIndex{98}$$

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

In the following exercises, multiply. Use either method.

Exercise $$\PageIndex{99}$$

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

$$w^{3}-16 w^{2}+73 w-70$$

Exercise $$\PageIndex{100}$$

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

Exercise $$\PageIndex{101}$$

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

$$3 q^{3}-11 q^{2}-19 q-5$$

Exercise $$\PageIndex{102}$$

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

Mixed Practice

Exercise $$\PageIndex{103}$$

(10y−6)+(4y−7)

14y−13

Exercise $$\PageIndex{104}$$

(15p−4)+(3p−5)

Exercise $$\PageIndex{105}$$

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

−11x−28

Exercise $$\PageIndex{106}$$

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

Exercise $$\PageIndex{107}$$

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

$$15 q^{3}-30 q^{2}+55 q$$

Exercise $$\PageIndex{108}$$

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

Exercise $$\PageIndex{109}$$

(s−7)(s+9)

$$s^{2}+2 s-63$$

Exercise $$\PageIndex{110}$$

(x−5)(x+13)

Exercise $$\PageIndex{111}$$

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

$$y^{3}-y^{2}-2 y$$

Exercise $$\PageIndex{112}$$

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

Exercise $$\PageIndex{113}$$

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

$$3 n^{3}-n^{2}-25 n+28$$

Exercise $$\PageIndex{114}$$

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

Exercise $$\PageIndex{115}$$

$$(7 p+10)(7 p-10)$$

$$49 p^{2}-100$$

Exercise $$\PageIndex{116}$$

(3y+8)(3y−8)

Exercise $$\PageIndex{117}$$

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

$$4 m^{4}-3 m^{3}-7 m^{2}$$

Exercise $$\PageIndex{118}$$

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

Exercise $$\PageIndex{119}$$

$$(5 a+7 b)(5 a+7 b)$$

$$25 a^{2}+70 a b+49 b^{2}$$

Exercise $$\PageIndex{120}$$

(3x−11y)(3x−11y)

Exercise $$\PageIndex{121}$$

(4y+12z)(4y−12z)

$$16 y^{2}-144 z^{2}$$

## 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?
1. 306
2. 306

## 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?

Exercise $$\PageIndex{126}$$

Multiply the following:

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

Exercise $$\PageIndex{127}$$

Multiply the following:

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

Exercise $$\PageIndex{128}$$

Multiply the following:

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

Exercise $$\PageIndex{129}$$

Multiply the following:

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