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Mathematics LibreTexts

6.E: Factoring (Exercises)

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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) 16b4 and 56b9

Answer

8b4

26) 28s2 and 36s4

27) 35z2 and 49z7

Answer

7z2

28) 24w3 and 30w8

29) 56x3y4 and 16x2y5

Answer

8x2y4

30) 35b5c3 and 63b4c4

31) 24s4t5 and 16s3t6

Answer

8s3t5

32) 10v4w3 and 8v3w4

33) 18y7, 45y6, and 27y5

Answer

9y5

34) 8r7, 24r6, and 12r5

35) 9a6, 6a5, and 15a4

Answer

3a4

36) 15a5, 24a4, and 24a3

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

37) 25a2+10a+20

Answer

5(5a2+2a+4)

38) 40c2+15c+40

39) 35s2+25s+45

Answer

5(7s2+5s+9)

40) 45b2+20b+35

41) 16c3+32c2+36c

Answer

4c(4c2+8c+9)

42) 12b3+12b2+18b

43) 42s3+24s2+18s

Answer

6s(7s2+4s+3)

44) 36y3+81y2+36y

45) 35s7+49s6+63s5

Answer

7s5(5s2+7s+9)

46) 35s7+56s6+56s5

47) 14b7+35b6+56b5

Answer

7b5(2b2+5b+8)

48) 45x5+81x4+45x3

49) 54y5z3+30y4z4+36y3z5

Answer

6y3z3(9y2+5yz+6z2)

50) 42x4y2+42x3y3+54x2y4

51) 45s4t3+40s3t4+15s2t5

Answer

5.82t3(9s2+8st+3t2)

52) 20v6w3+36v5w4+28v4w5

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

53) 7w(2w3)8(2w3)

Answer

(7w8)(2w3)

54) 5s(8s1)+4(8s1)

55) 9r(5r1)+8(5r1)

Answer

(9r+8)(5r1)

56) 5c(4c7)+2(4c7)

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

Answer

6(2a+5)(8a7)

58) 40v(7v4)+72(7v4)

59) 56a(2a1)21(2a1)

Answer

7(2a1)(8a3)

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

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

61) x2+2x9x18

Answer

(x9)(x+2)

62) x2+6x9x54

63) x2+3x+6x+18

Answer

(x+6)(x+3)

64) x2+8x+7x+56

65) x26x3x+18

Answer

(x3)(x6)

66) x23x9x+27

67) x29x+3x27

Answer

(x+3)(x9)

68) x22x+7x14

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

69) 8x2+3x56x21

Answer

(x7)(8x+3)

70) 4x2+9x32x72

71) 9x2+36x5x20

Answer

(9x5)(x+4)

72) 7x2+14x8x16

73) 6x27x48x+56

Answer

(x8)(6x7)

74) 8x27x72x+63

75) 2x2+12x+7x+42

Answer

(2x+7)(x+6)

76) 7x2+28x+9x+36

6.2: Solving Nonlinear Equations

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

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

Answer

x=29,38

2) (2x5)(7x4)=0

3) x(4x+7)(9x8)=0

Answer

x=0,74,89

4) x(9x8)(3x+1)=0

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

Answer

x=0,49

6) 4x(3x6)=0

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

Answer

x=1,6

8) (x4)(x1)=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) x2+7x=9x+63

Answer

Nonlinear

10) x2+9x=4x+36

11) 6x2=5x8

Answer

Linear

12) 5x+5=6x7

13) 7x2=2x

Answer

Nonlinear

14) 4x2=7x

15) 3x2+8x=9

Answer

Nonlinear

16) 5x22x=9

17) 3x+6=9

Answer

Linear

18) 8x5=3

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

19) 3x+8=9

Answer

13

20) 3x+4=2

21) 9x2=x

Answer

x=0,19

22) 6x2=7x

23) 3x+9=8x+7

Answer

25

24) 8x+5=6x+4

25) 8x2=2x

Answer

x=0,14

26) 8x2=18x

27) 9x+2=7

Answer

59

28) 3x+2=6

29) 9x2=6x

Answer

x=0,23

30) 6x2=14x

31) 7x2=4x

Answer

x=0,47

32) 7x2=9x

33) 7x+2=4x+7

Answer

53

34) 4x+3=2x+8

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

35) 63x2+56x+54x+48=0

Answer

x=67,89

36) 27x2+36x+6x+8=0

37) 16x218x+40x45=0

Answer

x=52,98

38) 42x235x+54x45=0

39) 45x2+18x+20x+8=0

Answer

x=49,25

40) 18x2+21x+30x+35=0

41) x2+10x+4x+40=0

Answer

x=4,10

42) x2+11x+10x+110=0

43) x2+6x11x66=0

Answer

x=11,6

44) x2+6x2x12=0

45) 15x224x+35x56=0

Answer

x=73,85

46) 12x210x+54x45=0

47) x2+2x+9x+18=0

Answer

x=9,2

48) x2+8x+4x+32=0

49) x2+4x8x32=0

Answer

x=8,4

50) x2+8x5x40=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) x2=4x

Answer

x=4,0

52) x2=6x

53) x2=5x

Answer

x=0,5

54) x2=6x

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) x2+7x=0

Answer

x=7,0

56) x28x=0

57) x23x=0

Answer

x=0,3

58) x2+2x=0

6.3: Factoring ax² + bx + c when a =1

In Exercises 1-6, compare the given trinomial with ax2+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) x2+7x18

Answer

(x2)(x+9)

2) x2+18x+80

3) x210x+9

Answer

(x1)(x9)

4) x2+12x+27

5) x2+14x+45

Answer

(x+5)(x+9)

6) x2+9x+20

In Exercises 7-12, compare the given trinomial with ax2+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) x216x+39

Answer

(x3)(x13)

8) x216x+48

9) x226x+69

Answer

(x3)(x23)

10) x222x+57

11) x225x+84

Answer

(x4)(x21)

12) x2+13x30

In Exercises 13-18, compare the given trinomial with ax2+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) x213x+36

Answer

(x4)(x9)

14) x2+x12

15) x2+10x+21

Answer

(x+3)(x+7)

16) x217x+66

17) x24x5

Answer

(x+1)(x5)

18) x220x+99

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

19) x2=7x+30

Answer

x=3,10

20) x2=2x+35

21) x2=11x10

Answer

x=1,10

22) x2=x+72

23) x2=15x50

Answer

x=5,10

24) x2=7x6

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

25) 60=x2+11x

Answer

x=4,15

26) 92=x227x

27) 11=x212x

Answer

x=1,11

28) 80=x216x

29) 56=x2+10x

Answer

x=4,14

30) 66=x2+19x

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

31) x2+20=12x

Answer

x=2,10

32) x212=11x

33) x236=9x

Answer

x=3,12

34) x2+6=5x

35) x2+8=6x

Answer

x=2,4

36) x2+77=18x

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) x2=x+12

Answer

x=3,4

38) x2=20x

39) x2+12=8x

Answer

x=2,6

40) x2+7=8x

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) x26x16=0

Answer

x=8,2

42) x2+7x18=0

43) x2+10x24=0

Answer

x=12,2

44) x29x36=0

6.4: Factoring ax² + bx + c when a≠1

In Exercises 1-6, compare the given trinomial with ax2+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) 6x2+13x5

Answer

(2x+5)(3x1)

2) 3x219x+20

3) 4x2x3

Answer

(x1)(4x+3)

4) 6x223x+7

5) 3x2+19x+28

Answer

(x+4)(3x+7)

6) 2x29x18

In Exercises 7-12, compare the given trinomial with ax2+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) 12x223x+5

Answer

(3x5)(4x1)

8) 8x2+22x+9

9) 6x2+17x+7

Answer

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

10) 4x2+19x+21

11) 3x2+4x32

Answer

(x+4)(3x8)

12) 4x2+x14

In Exercises 13-18, compare the given trinomial with ax2+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) 3x2+28x+9

Answer

(3x+1)(x+9)

14) 6x2+x1

15) 4x221x+5

Answer

(x5)(4x1)

16) 4x2x14

17) 6x211x7

Answer

(3x7)(2x+1)

18) 2x217x+21

In Exercises 19-26, factor the trinomial.

19) 16x536x4+14x3

Answer

2x3(2x1)(4x7)

20) 12x420x3+8x2

21) 36x475x3+21x2

Answer

3x2(3x1)(4x7)

22) 6x410x324x2

23) 6x433x3+42x2

Answer

3x2(x2)(2x7)

24) 15x310x2105x

25) 16x436x336x2

Answer

4x2(x3)(4x+3)

26) 40x410x35x2

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

27) (108)2(75)3

Answer

x=2,94

28) 2x2=7x3

29) 3x2+16=14x

Answer

x=2,83

30) 2x220=3x

31) 3x2+30=23x

Answer

x=6,53

32) 6x27=11x

33) 7x3=6x2

Answer

x=13,32

34) 13x45=2x2

35) 26x9=3x2

Answer

x=9,13

36) 23x+7=6x2

37) 6x2=25x+9

Answer

x=13,92

38) 2x2=13x+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) 2x29x5=0

Answer

x=12,5

40) 2x2+x28=0

41) 4x217x15=0

Answer

x=34,5

42) 3x2+14x24=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) 2x3=3x2+20x

Answer

x=0,52,4

44) 2x3=3x2+35x

45) 10x3+34x2=24x

Answer

x=0,4,35

46) 6x3+3x2=63x

6.5: Factoring Special Forms

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

1) (8r3t)2

Answer

64r248rt+9t2

2) (6a+c)2

3) (4a+7b)2

Answer

16a2+56ab+49b2

4) (4s+t)2

5) (s39)2

Answer

s618s3+81

6) (w3+7)2

7) (s2+6t2)2

Answer

s4+12s2t2+36t4

8) (7u22w2)2

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

9) 25s2+60st+36t2

Answer

(5s+6t)2

10) 9u2+24uv+16v2

11) 36v260vw+25w2

Answer

(6v5w)2

12) 49b242bc+9c2

13) a4+18a2b2+81b4

Answer

(a2+9b2)2

14) 64u4144u2w2+81w4

15) 49s428s2t2+4t4

Answer

(7s22t2)2

16) 4a412a2c2+9c4

17) 49b6112b3+64

Answer

(7b38)2

18) 25x610x3+1

19) 49r6+112r3+64

Answer

(7r3+8)2

20) a616a3+64

21) 5s3t20s2t2+20st3

Answer

5st(s2t)2

22) 12r3t12r2t2+3rt3

23) 8a3c+8a2c2+2ac3

Answer

2ac(2a+c)2

24) 18x3z60x2z2+50xz3

25) 48b3+120b275b

Answer

3b(4b5)2

26) 45c3+120c280c

27) 5u530u445u3

Answer

5u3(u+3)2

28) 12z536z427z3

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

29) (21c+16)(21c16)

Answer

441c2256

30) (19t+7)(19t7)

31) (5x+19z)(5x19z)

Answer

25x2361z2

32) (11u+5w)(11u5w)

33) (3y4+23z4)(3y423z4)

Answer

9y8529z8

34) (5x3+z3)(5x3z3)

35) (8r5+19s5)(8r519s5)

Answer

64r10361s10

36) (3u3+16v3)(3u316v3)

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

37) 361x2529

Answer

(19x+23)(19x23)

38) 9b225

39) 16v2169

Answer

(4v+13)(4v13)

40) 81r2169

41) 169x2576y2

Answer

(13x+24y)(13x24y)

42) 100y281z2

43) 529r2289s2

Answer

(23r+17s)(23r17s)

44) 49a2144b2

45) 49r6256t6

Answer

(7r3+16t3)(7r316t3)

46) 361x10484z10

47) 36u1025w10

Answer

(6u5+5w5)(6u55w5)

48) a681c6

49) 72y5242y3

Answer

2y3(6y+11)(6y11)

50) 75y5147y3

51) 1444a3b324ab3

Answer

4ab(19a+9b)(19a9b)

52) 12b3c1875bc3

53) 576x3z1156xz3

Answer

4xz(12x+17z)(12x17z)

54) 192u3v507uv3

55) 576t44t2

Answer

4t2(12t+1)(12t1)

56) 4z5256z3

57) 81x4256

Answer

(9x2+16)(3x+4)(3x4)

58) 81x41

59) 81x416

Answer

(9x2+4)(3x+2)(3x2)

60) x41

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

61) z3+z29z9

Answer

(z+3)(z3)(z+1)

62) 3u3+u248u16

63) x32x2yxy2+2y3

Answer

(x+y)(xy)(x2y)

64) x3+2x2z4xz28z3

65) r33r2t25rt2+75t3

Answer

(r+5t)(r5t)(r3t)

66) 2b33b2c50bc2+75c3

67) 2x3+x232x16

Answer

(x+4)(x4)(2x+1)

68) r32r2r+2

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

69) 2x3+7x2=72x+252

Answer

x=6,6,72

70) 2x3+7x2=32x+112

71) x3+5x2=64x+320

Answer

x=8,8,5

72) x3+4x2=49x+196

73) 144x2+121=264x

Answer

x=1112

74) 361x2+529=874x

75) 16x2=169

Answer

x=134,134

76) 289x2=4

77) 9x2=25

Answer

x=53,53

78) 144x2=121

79) 256x2+361=608x

Answer

x=1916

80) 16x2+289=136x

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) x3=x

Answer

x=0,1,1

82) x3=9x

83) 4x3=x

Answer

x=0,12,12

84) 9x3=x

6.6: Factoring Strategy

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

1) 484y4z2144y2z4

Answer

4y2z2(11y+6z)(11y6z)

2) 72s4t4242s2t6

3) 3x7z5363x5z5

Answer

3x5z5(x+11)(x11)

4) 5r5s280r3s2

5) 2u7162u5

Answer

2u5(u+9)(u9)

6) 405x4320x2

7) 3v81875v4

Answer

3v4(v2+25)(v+5)(v5)

8) 3a948a5

9) 3x6300x4

Answer

3x4(x+10)(x10)

10) 2y518y3

11) 1250u7w32u3w7

Answer

2u3w3(25u2+w2)(5u+w)(5uw)

12) 48y8z43y4z8

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

13) 75a6210a5+147a4

Answer

3a4(5a7)2

14) 245v7560v6+320v5

15) 180a5b3+540a4b4+405a3b5

Answer

45a3b3(2a+3b)2

16) 192u6v4+432u5v5+243u4v6

17) 2b5+4b4+2b3

Answer

2b3(b+1)2

18) 3v6+30v5+75v4

19) 2z44z3+2z2

Answer

2z2(z1)2

20) 2u640u5+200u4

21) 324x4+360x3+100x2

Answer

4x2(9x+5)2

22) 98b4+84b3+18b2

23) 75b4c5240b3c6+192b2c7

Answer

3b2c5(5b8c)2

24) 162a5c4180a4c5+50a3c6

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

25) 5a5+5a4210a3

Answer

5a3(a6)(a+7)

26) 3y59y412y3

27) 3y639y5+120y4

Answer

3y4(y8)(y5)

28) 3y727y6+42y5

29) 3z4+12z3135z2

Answer

3z2(z5)(z+9)

30) 5a440a345a2

31) 4a6+64a5+252a4

Answer

4a4(a+9)(a+7)

32) 4x4+64x3+252x2

33) 3z4+33z3+84z2

Answer

3z2(z+7)(z+4)

34) 5a6+65a5+180a4

35) 5z775z6+270z5

Answer

5z5(z6)(z9)

36) 3y427y3+24y2

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

37) 4b322b2+30b

Answer

2b(2b5)(b3)

38) 4b622b5+30b4

39) 2u4w53u3w620u2w7

Answer

u2w5(u4w)(2u+5w)

40) 12x5z2+9x4z330x3z4

41) 12x4y5+50x3y6+50x2y7

Answer

2x2y5(3x+5y)(2x+5y)

42) 24s4t3+62s3t4+40s2t5

43) 12x3+9x230x

Answer

3x(4x5)(x+2)

44) 6v4+2v320v2

45) 8u6+34u5+30u4

Answer

2u4(4u+5)(u+3)

46) 4a4+29a3+30a2

47) 12a4c435a3c5+25a2c6

Answer

a2cA(4a5c)(3a5c)

48) 18x6z539x5z6+18x4z7

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

49) 12y5+15y4108y3135y2

Answer

3y2(y+3)(y3)(4y+5)

50) 9b8+12b7324b6432b5

51) 9x6z5+6x5z6144x4z796x3z8

Answer

3x3z5(x+4z)(x4z)(3x+2z)

52) 12u7w3+9u6w4432u5w5324u4w6

53) 72z6+108z52z43z3

Answer

z3(6z+1)(6z1)(2z+3)

54) 216x7+324x66x59x4

55) 144a6c3+360a5c44a4c510a3c6

Answer

2a3c3(6a+c)(6ac)(2a+5c)

56) 48a8c4+32a7c53a6c62a5c7

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) 6x2+61x+120

Answer

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

58) 16x262x45

59) 60x2167x+72

Answer

(15x8)(4x9)

60) 28x2+x144

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=180tand the projectile’s height above sea level (in feet) is given by the equationy=16t2+352t+1664where 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=140tand the projectile’s height above sea level (in feet) is given by the equationy=16t2+288t+1408where 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 40π 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 180π 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 23. 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 34. 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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