Problem Solving

BTGmoderatorDC wrote:If H = [(x^3) - 6(x^2) - x + 30] / (x-5)and x ≠ 5, then H is equivalent to which of the following?

A. (x^2) - x - 6
B. (x^3) + 3(x^2) + 3x
C. (x^3) - 25
D. (x^3) - 5(x^2) - 3x
E. (x^2) + x + 10

OA A

Source: Manhattan Prep

Looking at the options, we can deduce that (x - 5) given in the denominator of H gets canceled.

Thus, (x - 5) must be a factor of the numerator of H = x^3 - 6x^2 - x + 30

x^3 - 6x^2 - x + 30
x^2(x - 5) - x^2 - x + 30
x^2(x - 5) - x(x - 5) - 6x + 30
x^2(x - 5) - x(x - 5) - 6(x - 5)

(x - 5)(x^2 - x - 6)

Thus,

H = (x^3 - 6x^2 - x + 30) / (x - 5)

= (x - 5)(x^2 - x - 6) / (x - 5)

= x^2 - x - 6; cancelling (x - 5)

The correct answer: A

Hope this helps!

-Jay
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BTGmoderatorDC wrote:If H = [(x^3) - 6(x^2) - x + 30] / (x-5)and x ≠ 5, then H is equivalent to which of the following?

A. (x^2) - x - 6
B. (x^3) + 3(x^2) + 3x
C. (x^3) - 25
D. (x^3) - 5(x^2) - 3x
E. (x^2) + x + 10

We see that the numerator of H, x^3 - 6x^2 - x + 30, is not a polynomial that can be easily factored. However, since x^3/x (the first term of numerator divided by the first term of the denominator) is x^2 and 30/(-5) (the constant term of numerator divided by the constant term of the denominator) is -6, we see that if H can be simplified, the answer must be choice A: x^2 - x - 6. Now let's verify it is the case by multiplying it by (x - 5) to see if the product is x^3 - 6x^2 - x + 30.

(x - 5)(x^2 - x - 6)

x^3 - x^2 - 6x - 5x^2 + 5x + 30

x^3 - 6x^2 - x + 30

We see that (x - 5)(x^2 - x - 6) = x^3 - 6x^2 - x + 30; therefore, (x^3 - 6x^2 - x + 30)/(x - 5) = x^2 - x - 6.

Answer: A