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Mathematics LibreTexts

6.E: Factoring (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

6.1: The Greatest Common Factor

In Exercises 1-6, list all positive divisors of the given number, in order, from smallest to largest.

1) 42

Answer

\{1,2,3,6,7,14,21,42\}

2) 60

3) 44

Answer

\{1,2,4,11,22,44\}

4) 85

5) 51

Answer

\{1,3,17,51\}

6) 63

In Exercises 7-12, list all common positive divisors of the given numbers, in order, from smallest to largest.

7) 36 and 42

Answer

\{1,2,3,6\}

8) 54 and 30

9) 78 and 54

Answer

\{1,2,3,6\}

10) 96 and 78

11) 8 and 76

Answer

\{1,2,4\}

12) 99 and 27

In Exercises 13-18, state the greatest common divisor of the given numbers.

13) 76 and 8

Answer

4

14) 84 and 60

15) 32 and 36

Answer

4

16) 64 and 76

17) 24 and 28

Answer

4

18) 63 and 27

In Exercises 19-24, use prime factorization to help calculate the greatest common divisor of the given numbers.

19) 600 and 1080

Answer

120

20) 150 and 120

21) 1800 and 2250

Answer

450

22) 540 and 150

23) 600 and 450

Answer

150

24) 4500 and 1800

In Exercises 25-36, find the greatest common factor of the given expressions.

25) 16b^{4} and 56b^{9}

Answer

8b^{4}

26) 28s^{2} and 36s^{4}

27) 35z^{2} and 49z^{7}

Answer

7z^{2}

28) 24w^{3} and 30w^{8}

29) 56x^{3} y^{4} and 16x^{2} y^{5}

Answer

8 x^{2} y^{4}

30) 35b^{5} c^{3} and 63b^{4} c^{4}

31) 24s^{4} t^{5} and 16s^{3} t^{6}

Answer

8 s^{3} t^{5}

32) 10v^{4} w^{3} and 8v^{3} w^{4}

33) 18 y^{7}, 45 y^{6}, and 27y^{5}

Answer

9 y^{5}

34) 8 r^{7}, 24 r^{6}, and 12r^{5}

35) 9 a^{6}, 6 a^{5}, and 15a^{4}

Answer

3 a^{4}

36) 15 a^{5}, 24 a^{4}, and 24a^{3}

In Exercises 37-52, factor out the \mathrm{GCF} from each of the given expressions.

37) 25 a^{2}+10 a+20

Answer

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

38) 40 c^{2}+15 c+40

39) 35 s^{2}+25 s+45

Answer

5\left(7 s^{2}+5 s+9\right)

40) 45 b^{2}+20 b+35

41) 16 c^{3}+32 c^{2}+36 c

Answer

4 c\left(4 c^{2}+8 c+9\right)

42) 12 b^{3}+12 b^{2}+18 b

43) 42 s^{3}+24 s^{2}+18 s

Answer

6 s\left(7 s^{2}+4 s+3\right)

44) 36 y^{3}+81 y^{2}+36 y

45) 35 s^{7}+49 s^{6}+63 s^{5}

Answer

7 s^{5}\left(5 s^{2}+7 s+9\right)

46) 35 s^{7}+56 s^{6}+56 s^{5}

47) 14 b^{7}+35 b^{6}+56 b^{5}

Answer

7 b^{5}\left(2 b^{2}+5 b+8\right)

48) 45 x^{5}+81 x^{4}+45 x^{3}

49) 54 y^{5} z^{3}+30 y^{4} z^{4}+36 y^{3} z^{5}

Answer

6 y^{3} z^{3}\left(9 y^{2}+5 y z+6 z^{2}\right)

50) 42 x^{4} y^{2}+42 x^{3} y^{3}+54 x^{2} y^{4}

51) 45 s^{4} t^{3}+40 s^{3} t^{4}+15 s^{2} t^{5}

Answer

5.8^{2} t^{3}\left(9 s^{2}+8 s t+3 t^{2}\right)

52) 20 v^{6} w^{3}+36 v^{5} w^{4}+28 v^{4} w^{5}

In Exercises 53-60, factor out the \mathrm{GCF} from each of the given expressions.

53) 7 w(2 w-3)-8(2 w-3)

Answer

(7 w-8)(2 w-3)

54) 5 s(8 s-1)+4(8 s-1)

55) 9 r(5 r-1)+8(5 r-1)

Answer

(9 r+8)(5 r-1)

56) 5 c(4 c-7)+2(4 c-7)

57) 48 a(2 a+5)-42(2 a+5)

Answer

6(2 a+5)(8 a-7)

58) 40 v(7 v-4)+72(7 v-4)

59) 56 a(2 a-1)-21(2 a-1)

Answer

7(2 a-1)(8 a-3)

60) 48 r(5 r+3)-40(5 r+3)

In Exercises 61-68, factor by grouping. Do not simplify the expression before factoring.

61) x^{2}+2 x-9 x-18

Answer

(x-9)(x+2)

62) x^{2}+6 x-9 x-54

63) x^{2}+3 x+6 x+18

Answer

(x+6)(x+3)

64) x^{2}+8 x+7 x+56

65) x^{2}-6 x-3 x+18

Answer

(x-3)(x-6)

66) x^{2}-3 x-9 x+27

67) x^{2}-9 x+3 x-27

Answer

(x+3)(x-9)

68) x^{2}-2 x+7 x-14

In Exercises 69-76, factor by grouping. Do not simplify the expression before factoring.

69) 8 x^{2}+3 x-56 x-21

Answer

(x-7)(8 x+3)

70) 4 x^{2}+9 x-32 x-72

71) 9 x^{2}+36 x-5 x-20

Answer

(9 x-5)(x+4)

72) 7 x^{2}+14 x-8 x-16

73) 6 x^{2}-7 x-48 x+56

Answer

(x-8)(6 x-7)

74) 8 x^{2}-7 x-72 x+63

75) 2 x^{2}+12 x+7 x+42

Answer

(2 x+7)(x+6)

76) 7 x^{2}+28 x+9 x+36

6.2: Solving Nonlinear Equations

In Exercises 1-8, solve the given equation for x.

1) (9 x+2)(8 x+3)=0

Answer

x=-\dfrac{2}{9},-\dfrac{3}{8}

2) (2 x-5)(7 x-4)=0

3) x(4 x+7)(9 x-8)=0

Answer

x=0,-\dfrac{7}{4}, \dfrac{8}{9}

4) x(9 x-8)(3 x+1)=0

5) -9 x(9 x+4)=0

Answer

x=0,-\dfrac{4}{9}

6) 4 x(3 x-6)=0

7) (x+1)(x+6)=0

Answer

x=-1,-6

8) (x-4)(x-1)=0

In Exercises 9-18, given that you are solving for x, state whether the given equation is linear or nonlinear. Do not solve the equation.

9) x^{2}+7 x=9 x+63

Answer

Nonlinear

10) x^{2}+9 x=4 x+36

11) 6 x-2=5 x-8

Answer

Linear

12) -5 x+5=-6 x-7

13) 7 x^{2}=-2 x

Answer

Nonlinear

14) 4 x^{2}=-7 x

15) 3 x^{2}+8 x=-9

Answer

Nonlinear

16) 5 x^{2}-2 x=-9

17) -3 x+6=-9

Answer

Linear

18) 8 x-5=3

In Exercises 19-34, solve each of the given equations for x.

19) 3 x+8=9

Answer

\dfrac{1}{3}

20) 3 x+4=2

21) 9 x^{2}=-x

Answer

x=0,-\dfrac{1}{9}

22) 6 x^{2}=7 x

23) 3 x+9=8 x+7

Answer

\dfrac{2}{5}

24) 8 x+5=6 x+4

25) 8 x^{2}=-2 x

Answer

x=0,-\dfrac{1}{4}

26) 8 x^{2}=18 x

27) 9 x+2=7

Answer

\dfrac{5}{9}

28) 3 x+2=6

29) 9 x^{2}=6 x

Answer

x=0, \dfrac{2}{3}

30) 6 x^{2}=-14 x

31) 7 x^{2}=-4 x

Answer

x=0,-\dfrac{4}{7}

32) 7 x^{2}=-9 x

33) 7 x+2=4 x+7

Answer

\dfrac{5}{3}

34) 4 x+3=2 x+8

In Exercises 35-50, factor by grouping to solve each of the given equations for x.

35) 63 x^{2}+56 x+54 x+48=0

Answer

x=-\dfrac{6}{7},-\dfrac{8}{9}

36) 27 x^{2}+36 x+6 x+8=0

37) 16 x^{2}-18 x+40 x-45=0

Answer

x=-\dfrac{5}{2}, \dfrac{9}{8}

38) 42 x^{2}-35 x+54 x-45=0

39) 45 x^{2}+18 x+20 x+8=0

Answer

x=-\dfrac{4}{9},-\dfrac{2}{5}

40) 18 x^{2}+21 x+30 x+35=0

41) x^{2}+10 x+4 x+40=0

Answer

x=-4,-10

42) x^{2}+11 x+10 x+110=0

43) x^{2}+6 x-11 x-66=0

Answer

x=11,-6

44) x^{2}+6 x-2 x-12=0

45) 15 x^{2}-24 x+35 x-56=0

Answer

x=-\dfrac{7}{3}, \dfrac{8}{5}

46) 12 x^{2}-10 x+54 x-45=0

47) x^{2}+2 x+9 x+18=0

Answer

x=-9,-2

48) x^{2}+8 x+4 x+32=0

49) x^{2}+4 x-8 x-32=0

Answer

x=8,-4

50) x^{2}+8 x-5 x-40=0

In Exercises 51-54, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 5:intersect utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.2.7.

51) x^{2}=-4 x

Answer

x=-4,0

52) x^{2}=6 x

53) x^{2}=5 x

Answer

x=0,5

54) x^{2}=-6 x

In Exercises 55-58, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 2:zero utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.2.8.

55) x^{2}+7 x=0

Answer

x=-7,0

56) x^{2}-8 x=0

57) x^{2}-3 x=0

Answer

x=0,3

58) x^{2}+2 x=0

6.3: Factoring ax² + bx + c when a =1

In Exercises 1-6, compare the given trinomial with ax^2 + bx + c, then list ALL integer pairs whose product equals ac. Circle the pair whose sum equals b, then use this pair to help factor the given trinomial.

1) x^{2}+7 x-18

Answer

(x-2)(x+9)

2) x^{2}+18 x+80

3) x^{2}-10 x+9

Answer

(x-1)(x-9)

4) x^{2}+12 x+27

5) x^{2}+14 x+45

Answer

(x+5)(x+9)

6) x^{2}+9 x+20

In Exercises 7-12, compare the given trinomial with ax^2 +bx+c, then begin listing integer pairs whose product equals ac. Cease the list process when you discover a pair whose sum equals b, then circle and use this pair to help factor the given trinomial.

7) x^{2}-16 x+39

Answer

(x-3)(x-13)

8) x^{2}-16 x+48

9) x^{2}-26 x+69

Answer

(x-3)(x-23)

10) x^{2}-22 x+57

11) x^{2}-25 x+84

Answer

(x-4)(x-21)

12) x^{2}+13 x-30

In Exercises 13-18, compare the given trinomial with ax^2 + bx + c, then compute ac. Try to mentally discover the integer pair whose product is ac and whose sum is b. Factor the trinomial by “dropping this pair in place.”

Note: If you find you cannot identify the pair mentally, begin listing integer pairs whose product equals ac, then cease the listing process when you encounter the pair whose sum equals b.

13) x^{2}-13 x+36

Answer

(x-4)(x-9)

14) x^{2}+x-12

15) x^{2}+10 x+21

Answer

(x+3)(x+7)

16) x^{2}-17 x+66

17) x^{2}-4 x-5

Answer

(x+1)(x-5)

18) x^{2}-20 x+99

In Exercises 19-24, use an algebraic technique to solve the given equation.

19) x^{2}=-7 x+30

Answer

x=3,-10

20) x^{2}=-2 x+35

21) x^{2}=-11 x-10

Answer

x=-1,-10

22) x^{2}=x+72

23) x^{2}=-15 x-50

Answer

x=-5,-10

24) x^{2}=-7 x-6

In Exercises 25-30, use an algebraic technique to solve the given equation.

25) 60=x^{2}+11 x

Answer

x=4,-15

26) -92=x^{2}-27 x

27) -11=x^{2}-12 x

Answer

x=1,11

28) 80=x^{2}-16 x

29) 56=x^{2}+10 x

Answer

x=4,-14

30) 66=x^{2}+19 x

In Exercises 31-36, use an algebraic technique to solve the given equation.

31) x^{2}+20=-12 x

Answer

x=-2,-10

32) x^{2}-12=11 x

33) x^{2}-36=9 x

Answer

x=-3,12

34) x^{2}+6=5 x

35) x^{2}+8=-6 x

Answer

x=-2,-4

36) x^{2}+77=18 x

In Exercises 37-40, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 5:intersect utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.3.6.

37) x^{2}=x+12

Answer

x=-3,4

38) x^{2}=20-x

39) x^{2}+12=8 x

Answer

x=2,6

40) x^{2}+7=8 x

In Exercises 41-44, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 2:zero utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.3.7.

41) x^{2}-6 x-16=0

Answer

x=8,-2

42) x^{2}+7 x-18=0

43) x^{2}+10 x-24=0

Answer

x=-12,2

44) x^{2}-9 x-36=0

6.4: Factoring ax² + bx + c when a≠1

In Exercises 1-6, compare the given trinomial with ax^2 + bx + c, then list ALL integer pairs whose product equals ac. Circle the pair whose sum equals b, then use this pair to help factor the given trinomial.

1) 6 x^{2}+13 x-5

Answer

(2 x+5)(3 x-1)

2) 3 x^{2}-19 x+20

3) 4 x^{2}-x-3

Answer

(x-1)(4 x+3)

4) 6 x^{2}-23 x+7

5) 3 x^{2}+19 x+28

Answer

(x+4)(3 x+7)

6) 2 x^{2}-9 x-18

In Exercises 7-12, compare the given trinomial with ax^2 +bx+c, then begin listing integer pairs whose product equals ac. Cease the list process when you discover a pair whose sum equals b, then circle and use this pair to help factor the given trinomial.

7) 12 x^{2}-23 x+5

Answer

(3 x-5)(4 x-1)

8) 8 x^{2}+22 x+9

9) 6 x^{2}+17 x+7

Answer

(2 x+1)(3 x+7)

10) 4 x^{2}+19 x+21

11) 3 x^{2}+4 x-32

Answer

(x+4)(3 x-8)

12) 4 x^{2}+x-14

In Exercises 13-18, compare the given trinomial with ax^2 + bx + c, then compute ac. Try to mentally discover the integer pair whose product is ac and whose sum is b. Use this pair to help factor the given trinomial.

Note: If you find you cannot identify the pair mentally, begin listing integer pairs whose product equals ac, then cease the listing process when you encounter the pair whose sum equals b.

13) 3 x^{2}+28 x+9

Answer

(3 x+1)(x+9)

14) 6 x^{2}+x-1

15) 4 x^{2}-21 x+5

Answer

(x-5)(4 x-1)

16) 4 x^{2}-x-14

17) 6 x^{2}-11 x-7

Answer

(3 x-7)(2 x+1)

18) 2 x^{2}-17 x+21

In Exercises 19-26, factor the trinomial.

19) 16 x^{5}-36 x^{4}+14 x^{3}

Answer

2 x^{3}(2 x-1)(4 x-7)

20) 12 x^{4}-20 x^{3}+8 x^{2}

21) 36 x^{4}-75 x^{3}+21 x^{2}

Answer

3 x^{2}(3 x-1)(4 x-7)

22) 6 x^{4}-10 x^{3}-24 x^{2}

23) 6 x^{4}-33 x^{3}+42 x^{2}

Answer

3 x^{2}(x-2)(2 x-7)

24) 15 x^{3}-10 x^{2}-105 x

25) 16 x^{4}-36 x^{3}-36 x^{2}

Answer

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

26) 40 x^{4}-10 x^{3}-5 x^{2}

In Exercises 27-38, use an algebraic technique to solve the given equation.

27) (10-8)^{2}-(7-5)^{3}

Answer

x=2,-\dfrac{9}{4}

28) 2 x^{2}=7 x-3

29) 3 x^{2}+16=-14 x

Answer

x=-2,-\dfrac{8}{3}

30) 2 x^{2}-20=-3 x

31) 3 x^{2}+30=23 x

Answer

x=6, \dfrac{5}{3}

32) 6 x^{2}-7=-11 x

33) -7 x-3=-6 x^{2}

Answer

x=-\dfrac{1}{3}, \dfrac{3}{2}

34) 13 x-45=-2 x^{2}

35) 26 x-9=-3 x^{2}

Answer

x=-9, \dfrac{1}{3}

36) -23 x+7=-6 x^{2}

37) 6 x^{2}=-25 x+9

Answer

x=\dfrac{1}{3},-\dfrac{9}{2}

38) 2 x^{2}=13 x+45

In Exercises 39-42, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 2:zero utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.4.5.

39) 2 x^{2}-9 x-5=0

Answer

x=-\dfrac{1}{2},5

40) 2 x^{2}+x-28=0

41) 4 x^{2}-17 x-15=0

Answer

x=-\dfrac{3}{4},5

42) 3 x^{2}+14 x-24=0

In Exercises 43-46, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 2:zero utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.4.6.

43) 2 x^{3}=3 x^{2}+20 x

Answer

x=0,-\dfrac{5}{2},4

44) 2 x^{3}=3 x^{2}+35 x

45) 10 x^{3}+34 x^{2}=24 x

Answer

x=0,-4, \dfrac{3}{5}

46) 6 x^{3}+3 x^{2}=63 x

6.5: Factoring Special Forms

In Exercises 1-8, expand each of the given expressions.

1) (8 r-3 t)^{2}

Answer

64 r^{2}-48 r t+9 t^{2}

2) (6 a+c)^{2}

3) (4 a+7 b)^{2}

Answer

16 a^{2}+56 a b+49 b^{2}

4) (4 s+t)^{2}

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

Answer

s^{6}-18 s^{3}+81

6) \left(w^{3}+7\right)^{2}

7) \left(s^{2}+6 t^{2}\right)^{2}

Answer

s^{4}+12 s^{2} t^{2}+36 t^{4}

8) \left(7 u^{2}-2 w^{2}\right)^{2}

In Exercises 9-28, factor each of the given expressions.

9) 25 s^{2}+60 s t+36 t^{2}

Answer

(5 s+6 t)^{2}

10) 9 u^{2}+24 u v+16 v^{2}

11) 36 v^{2}-60 v w+25 w^{2}

Answer

(6 v-5 w)^{2}

12) 49 b^{2}-42 b c+9 c^{2}

13) a^{4}+18 a^{2} b^{2}+81 b^{4}

Answer

\left(a^{2}+9 b^{2}\right)^{2}

14) 64 u^{4}-144 u^{2} w^{2}+81 w^{4}

15) 49 s^{4}-28 s^{2} t^{2}+4 t^{4}

Answer

\left(7 s^{2}-2 t^{2}\right)^{2}

16) 4 a^{4}-12 a^{2} c^{2}+9 c^{4}

17) 49 b^{6}-112 b^{3}+64

Answer

\left(7 b^{3}-8\right)^{2}

18) 25 x^{6}-10 x^{3}+1

19) 49 r^{6}+112 r^{3}+64

Answer

\left(7 r^{3}+8\right)^{2}

20) a^{6}-16 a^{3}+64

21) 5 s^{3} t-20 s^{2} t^{2}+20 s t^{3}

Answer

5 s t(s-2 t)^{2}

22) 12 r^{3} t-12 r^{2} t^{2}+3 r t^{3}

23) 8 a^{3} c+8 a^{2} c^{2}+2 a c^{3}

Answer

2 a c(2 a+c)^{2}

24) 18 x^{3} z-60 x^{2} z^{2}+50 x z^{3}

25) -48 b^{3}+120 b^{2}-75 b

Answer

-3 b(4 b-5)^{2}

26) -45 c^{3}+120 c^{2}-80 c

27) -5 u^{5}-30 u^{4}-45 u^{3}

Answer

-5 u^{3}(u+3)^{2}

28) -12 z^{5}-36 z^{4}-27 z^{3}

In Exercises 29-36, expand each of the given expressions.

29) (21 c+16)(21 c-16)

Answer

441 c^{2}-256

30) (19 t+7)(19 t-7)

31) (5 x+19 z)(5 x-19 z)

Answer

25 x^{2}-361 z^{2}

32) (11 u+5 w)(11 u-5 w)

33) \left(3 y^{4}+23 z^{4}\right)\left(3 y^{4}-23 z^{4}\right)

Answer

9 y^{8}-529 z^{8}

34) \left(5 x^{3}+z^{3}\right)\left(5 x^{3}-z^{3}\right)

35) \left(8 r^{5}+19 s^{5}\right)\left(8 r^{5}-19 s^{5}\right)

Answer

64 r^{10}-361 s^{10}

36) \left(3 u^{3}+16 v^{3}\right)\left(3 u^{3}-16 v^{3}\right)

In Exercises 37-60, factor each of the given expressions.

37) 361 x^{2}-529

Answer

(19 x+23)(19 x-23)

38) 9 b^{2}-25

39) 16 v^{2}-169

Answer

(4 v+13)(4 v-13)

40) 81 r^{2}-169

41) 169 x^{2}-576 y^{2}

Answer

(13 x+24 y)(13 x-24 y)

42) 100 y^{2}-81 z^{2}

43) 529 r^{2}-289 s^{2}

Answer

(23 r+17 s)(23 r-17 s)

44) 49 a^{2}-144 b^{2}

45) 49 r^{6}-256 t^{6}

Answer

\left(7 r^{3}+16 t^{3}\right)\left(7 r^{3}-16 t^{3}\right)

46) 361 x^{10}-484 z^{10}

47) 36 u^{10}-25 w^{10}

Answer

\left(6 u^{5}+5 w^{5}\right)\left(6 u^{5}-5 w^{5}\right)

48) a^{6}-81 c^{6}

49) 72 y^{5}-242 y^{3}

Answer

2 y^{3}(6 y+11)(6 y-11)

50) 75 y^{5}-147 y^{3}

51) 1444 a^{3} b-324 a b^{3}

Answer

4 a b(19 a+9 b)(19 a-9 b)

52) 12 b^{3} c-1875 b c^{3}

53) 576 x^{3} z-1156 x z^{3}

Answer

4 x z(12 x+17 z)(12 x-17 z)

54) 192 u^{3} v-507 u v^{3}

55) 576 t^{4}-4 t^{2}

Answer

4 t^{2}(12 t+1)(12 t-1)

56) 4 z^{5}-256 z^{3}

57) 81 x^{4}-256

Answer

\left(9 x^{2}+16\right)(3 x+4)(3 x-4)

58) 81 x^{4}-1

59) 81 x^{4}-16

Answer

\left(9 x^{2}+4\right)(3 x+2)(3 x-2)

60) x^{4}-1

In Exercises 61-68, factor each of the given expressions completely.

61) z^{3}+z^{2}-9 z-9

Answer

(z+3)(z-3)(z+1)

62) 3 u^{3}+u^{2}-48 u-16

63) x^{3}-2 x^{2} y-x y^{2}+2 y^{3}

Answer

(x+y)(x-y)(x-2 y)

64) x^{3}+2 x^{2} z-4 x z^{2}-8 z^{3}

65) r^{3}-3 r^{2} t-25 r t^{2}+75 t^{3}

Answer

(r+5 t)(r-5 t)(r-3 t)

66) 2 b^{3}-3 b^{2} c-50 b c^{2}+75 c^{3}

67) 2 x^{3}+x^{2}-32 x-16

Answer

(x+4)(x-4)(2 x+1)

68) r^{3}-2 r^{2}-r+2

In Exercises 69-80, solve each of the given equations for x.

69) 2 x^{3}+7 x^{2}=72 x+252

Answer

x=-6,6,-\dfrac{7}{2}

70) 2 x^{3}+7 x^{2}=32 x+112

71) x^{3}+5 x^{2}=64 x+320

Answer

x=-8,8,-5

72) x^{3}+4 x^{2}=49 x+196

73) 144 x^{2}+121=264 x

Answer

x=\dfrac{11}{12}

74) 361 x^{2}+529=874 x

75) 16 x^{2}=169

Answer

x=-\dfrac{13}{4}, \dfrac{13}{4}

76) 289 x^{2}=4

77) 9 x^{2}=25

Answer

x=-\dfrac{5}{3}, \dfrac{5}{3}

78) 144 x^{2}=121

79) 256 x^{2}+361=-608 x

Answer

x=-\dfrac{19}{16}

80) 16 x^{2}+289=-136 x

In Exercises 81-84, perform each of the following tasks:

  1. Use a strictly algebraic technique to solve the given equation.
  2. Use the 5:intersect utility on your graphing calculator to solve the given equation.

Report the results found using graphing calculator as shown in Example 6.5.12.

81) x^{3}=x

Answer

x=0,-1,1

82) x^{3}=9 x

83) 4 x^{3}=x

Answer

x=0,-\dfrac{1}{2},\dfrac{1}{2}

84) 9 x^{3}=x

6.6: Factoring Strategy

In Exercises 1-12, factor each of the given polynomials completely.

1) 484 y^{4} z^{2}-144 y^{2} z^{4}

Answer

4 y^{2} z^{2}(11 y+6 z)(11 y-6 z)

2) 72 s^{4} t^{4}-242 s^{2} t^{6}

3) 3 x^{7} z^{5}-363 x^{5} z^{5}

Answer

3 x^{5} z^{5}(x+11)(x-11)

4) 5 r^{5} s^{2}-80 r^{3} s^{2}

5) 2 u^{7}-162 u^{5}

Answer

2u^5(u + 9)(u−9)

6) 405 x^{4}-320 x^{2}

7) 3 v^{8}-1875 v^{4}

Answer

3v^4(v^2 + 25)(v + 5)(v−5)

8) 3 a^{9}-48 a^{5}

9) 3 x^{6}-300 x^{4}

Answer

3 x^{4}(x+10)(x-10)

10) 2 y^{5}-18 y^{3}

11) 1250 u^{7} w^{3}-2 u^{3} w^{7}

Answer

2 u^{3} w^{3}\left(25 u^{2}+w^{2}\right)(5 u+w)(5 u-w)

12) 48 y^{8} z^{4}-3 y^{4} z^{8}

In Exercises 13-24, factor each of the given polynomials completely.

13) 75 a^{6}-210 a^{5}+147 a^{4}

Answer

3 a^{4}(5 a-7)^{2}

14) 245 v^{7}-560 v^{6}+320 v^{5}

15) 180 a^{5} b^{3}+540 a^{4} b^{4}+405 a^{3} b^{5}

Answer

45 a^{3} b^{3}(2 a+3 b)^{2}

16) 192 u^{6} v^{4}+432 u^{5} v^{5}+243 u^{4} v^{6}

17) 2 b^{5}+4 b^{4}+2 b^{3}

Answer

2 b^{3}(b+1)^{2}

18) 3 v^{6}+30 v^{5}+75 v^{4}

19) 2 z^{4}-4 z^{3}+2 z^{2}

Answer

2 z^{2}(z-1)^{2}

20) 2 u^{6}-40 u^{5}+200 u^{4}

21) 324 x^{4}+360 x^{3}+100 x^{2}

Answer

4 x^{2}(9 x+5)^{2}

22) 98 b^{4}+84 b^{3}+18 b^{2}

23) 75 b^{4} c^{5}-240 b^{3} c^{6}+192 b^{2} c^{7}

Answer

3 b^{2} c^{5}(5 b-8 c)^{2}

24) 162 a^{5} c^{4}-180 a^{4} c^{5}+50 a^{3} c^{6}

In Exercises 25-36, factor each of the given polynomials completely.

25) 5 a^{5}+5 a^{4}-210 a^{3}

Answer

5 a^{3}(a-6)(a+7)

26) 3 y^{5}-9 y^{4}-12 y^{3}

27) 3 y^{6}-39 y^{5}+120 y^{4}

Answer

3 y^{4}(y-8)(y-5)

28) 3 y^{7}-27 y^{6}+42 y^{5}

29) 3 z^{4}+12 z^{3}-135 z^{2}

Answer

3 z^{2}(z-5)(z+9)

30) 5 a^{4}-40 a^{3}-45 a^{2}

31) 4 a^{6}+64 a^{5}+252 a^{4}

Answer

4 a^{4}(a+9)(a+7)

32) 4 x^{4}+64 x^{3}+252 x^{2}

33) 3 z^{4}+33 z^{3}+84 z^{2}

Answer

3 z^{2}(z+7)(z+4)

34) 5 a^{6}+65 a^{5}+180 a^{4}

35) 5 z^{7}-75 z^{6}+270 z^{5}

Answer

5 z^{5}(z-6)(z-9)

36) 3 y^{4}-27 y^{3}+24 y^{2}

In Exercises 37-48, factor each of the given polynomials completely.

37) 4 b^{3}-22 b^{2}+30 b

Answer

2 b(2 b-5)(b-3)

38) 4 b^{6}-22 b^{5}+30 b^{4}

39) 2 u^{4} w^{5}-3 u^{3} w^{6}-20 u^{2} w^{7}

Answer

u^{2} w^{5}(u-4 w)(2 u+5 w)

40) 12 x^{5} z^{2}+9 x^{4} z^{3}-30 x^{3} z^{4}

41) 12 x^{4} y^{5}+50 x^{3} y^{6}+50 x^{2} y^{7}

Answer

2 x^{2} y^{5}(3 x+5 y)(2 x+5 y)

42) 24 s^{4} t^{3}+62 s^{3} t^{4}+40 s^{2} t^{5}

43) 12 x^{3}+9 x^{2}-30 x

Answer

3 x(4 x-5)(x+2)

44) 6 v^{4}+2 v^{3}-20 v^{2}

45) 8 u^{6}+34 u^{5}+30 u^{4}

Answer

2 u^{4}(4 u+5)(u+3)

46) 4 a^{4}+29 a^{3}+30 a^{2}

47) 12 a^{4} c^{4}-35 a^{3} c^{5}+25 a^{2} c^{6}

Answer

a^{2} c^{A}(4 a-5 c)(3 a-5 c)

48) 18 x^{6} z^{5}-39 x^{5} z^{6}+18 x^{4} z^{7}

In Exercises 49-56, factor each of the given polynomials completely.

49) 12 y^{5}+15 y^{4}-108 y^{3}-135 y^{2}

Answer

3 y^{2}(y+3)(y-3)(4 y+5)

50) 9 b^{8}+12 b^{7}-324 b^{6}-432 b^{5}

51) 9 x^{6} z^{5}+6 x^{5} z^{6}-144 x^{4} z^{7}-96 x^{3} z^{8}

Answer

3 x^{3} z^{5}(x+4 z)(x-4 z)(3 x+2 z)

52) 12 u^{7} w^{3}+9 u^{6} w^{4}-432 u^{5} w^{5}-324 u^{4} w^{6}

53) 72 z^{6}+108 z^{5}-2 z^{4}-3 z^{3}

Answer

z^{3}(6 z+1)(6 z-1)(2 z+3)

54) 216 x^{7}+324 x^{6}-6 x^{5}-9 x^{4}

55) 144 a^{6} c^{3}+360 a^{5} c^{4}-4 a^{4} c^{5}-10 a^{3} c^{6}

Answer

2 a^{3} c^{3}(6 a+c)(6 a-c)(2 a+5 c)

56) 48 a^{8} c^{4}+32 a^{7} c^{5}-3 a^{6} c^{6}-2 a^{5} c^{7}

In Exercises 57-60, use your calculator to help factor each of the given trinomials. Follow the procedure outline in Using the Calculator to Assist the ac-Method.

57) 6 x^{2}+61 x+120

Answer

(2 x+15)(3 x+8)

58) 16 x^{2}-62 x-45

59) 60 x^{2}-167 x+72

Answer

(15 x-8)(4 x-9)

60) 28 x^{2}+x-144

6.7: Applications of Factoring

1) A rectangular canvas picture measures 14 inches by 36 inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to 720 square inches. Find the uniform width of the frame.

Answer

2 inches

2) A rectangular canvas picture measures 10 inches by 32 inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to 504 square inches. Find the uniform width of the frame.

3) A projectile is fired at an angle into the air from atop a cliff overlooking the ocean. The projectile’s distance (in feet) from the base of the cliff is give by the equationx = 180t \nonumber and the projectile’s height above sea level (in feet) is given by the equationy = −16t^2 + 352t + 1664 \nonumber where t is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliff?

Answer

26 seconds, 4,680 feet

4) A projectile is fired at an angle into the air from atop a cliff overlooking the ocean. The projectile’s distance (in feet) from the base of the cliff is give by the equationx = 140t \nonumber and the projectile’s height above sea level (in feet) is given by the equationy = −16t^2 + 288t + 1408 \nonumber where t is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliff?

5) The product of two consecutive even integers is 624. Find the integers.

Answer

-26 and −24, and 24 and 26

6) The product of two consecutive even integers is 528. Find the integers.

7) The product of two consecutive positive integers is 552. Find the integers.

Answer

23, 24

8) The product of two consecutive positive integers is 756. Find the integers.

9) The product of two consecutive odd integers is 483. Find the integers.

Answer

−23 and −21, and 21 and 23

10) The product of two consecutive odd integers is 783. Find the integers.

11) A rectangle has perimeter 42 feet and area 104 square feet. Find the dimensions of the rectangle.

Answer

8 feet by 13 feet

12) A rectangle has perimeter 32 feet and area 55 square feet. Find the dimensions of the rectangle.

13) The radius of the outer circle is one inch longer than twice the radius of the inner circle.

Exercise 6.7.13_14.png

If the area of the shaded region is 40\pi square inches, what is the length of the inner radius?

Answer

3 inches

14) The radius of the outer circle is two inches longer than three times the radius of the inner circle.

Exercise 6.7.13_14.png

If the area of the shaded region is 180\pi square inches, what is the length of the inner radius?

15) You have two positive numbers. The second number is three more than two times the first number. The difference of their squares is 144. Find both positive numbers.

Answer

5 and 13

16) You have two positive numbers. The second number is two more than three times the first number. The difference of their squares is 60. Find both positive numbers.

17) Two numbers differ by 5. The sum of their squares is 97. Find the two numbers.

Answer

4 and 9, and −4 and −9

18) Two numbers differ by 6. The sum of their squares is 146. Find the two numbers.

19) The length of a rectangle is three feet longer than six times its width. If the area of the rectangle is 165 square feet, what is the width of the rectangle?

Answer

5 feet

20) The length of a rectangle is three feet longer than nine times its width. If the area of the rectangle is 90 square feet, what is the width of the rectangle?

21) The ratio of the width to the length of a given rectangle is 2 to 3, or \dfrac {2}{3}. If the width and length are both increased by 4 inches, the area of the resulting rectangle is 80 square inches. Find the width and length of the original rectangle.

Answer

4 inches by 6 inches

22) The ratio of the width to the length of a given rectangle is 3 to 4, or \dfrac {3}{4}. If the width is increased by 3 inches and the length is increased by 6 inches, the area of the resulting rectangle is 126 square inches. Find the width and length of the original rectangle.


This page titled 6.E: Factoring (Exercises) is shared under a CC BY-NC-ND 3.0 license and was authored, remixed, and/or curated by David Arnold via source content that was edited to the style and standards of the LibreTexts platform.

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