Answer:
1. (x - 16)/(x - 8)
2. (x + 12)/(x (x - 7))
3. (2 y (y^2 - 6))/(y^2 + 5)
4. (2 x (2 x^2 + 3))/(x^2 - 3)
Explanation:
Simplify the following:
(x^2 - 14 x - 32)/(x^2 - 6 x - 16)
The factors of -16 that sum to -6 are 2 and -8. So, x^2 - 6 x - 16 = (x + 2) (x - 8):
(x^2 - 14 x - 32)/((x + 2) (x - 8))
The factors of -32 that sum to -14 are 2 and -16. So, x^2 - 14 x - 32 = (x + 2) (x - 16):
((x + 2) (x - 16))/((x + 2) (x - 8))
((x + 2) (x - 16))/((x + 2) (x - 8)) = (x + 2)/(x + 2)×(x - 16)/(x - 8) = (x - 16)/(x - 8):
Answer: (x - 16)/(x - 8)
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Simplify the following:
(x^2 + 7 x - 60)/(x^3 - 12 x^2 + 35 x)
The factors of -60 that sum to 7 are 12 and -5. So, x^2 + 7 x - 60 = (x + 12) (x - 5):
((x + 12) (x - 5))/(x^3 - 12 x^2 + 35 x)
Factor x out of x^3 - 12 x^2 + 35 x:
((x + 12) (x - 5))/(x (x^2 - 12 x + 35))
The factors of 35 that sum to -12 are -5 and -7. So, x^2 - 12 x + 35 = (x - 5) (x - 7):
((x + 12) (x - 5))/(x (x - 5) (x - 7))
((x + 12) (x - 5))/(x (x - 5) (x - 7)) = (x - 5)/(x - 5)×(x + 12)/(x (x - 7)) = (x + 12)/(x (x - 7)):
Answer: (x + 12)/(x (x - 7))
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Simplify the following:
(6 y^3 - 36 y)/(3 y^2 + 15)
Factor 3 out of 3 y^2 + 15:
(6 y^3 - 36 y)/(3 (y^2 + 5))
Factor 6 y out of 6 y^3 - 36 y:
(6 y (y^2 - 6))/(3 (y^2 + 5))
6/3 = (3×2)/3 = 2:
Answer: (2 y (y^2 - 6))/(y^2 + 5)
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Simplify the following:
(20 x^4 + 30 x^2)/(5 x^3 - 15 x)
Factor 5 x out of 5 x^3 - 15 x:
(20 x^4 + 30 x^2)/(5 x (x^2 - 3))
Factor 10 x^2 out of 20 x^4 + 30 x^2:
(10 x^2 (2 x^2 + 3))/(5 x (x^2 - 3))
10/5 = (5×2)/5 = 2:
(2 x^2 (2 x^2 + 3))/(x (x^2 - 3))
Combine powers. (2 x^2 (2 x^2 + 3))/(x (x^2 - 3)) = (2 x^(2 - 1) (2 x^2 + 3))/(x^2 - 3):
(2 x^(2 - 1) (2 x^2 + 3))/(x^2 - 3)
2 - 1 = 1:
Answer: (2 x (2 x^2 + 3))/(x^2 - 3)