Renee is asked to find a factor of x^3−5x^2+6x, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)

Question

A. x−2

B. x+3

C. x+2

D. x^2+5x+6

Answer 1

To find a factor of x^3-5x^2+6x, we can plug in the given value for x and check if the expression equals zero.

For option A, x-2:
(x-2)^3 - 5(x-2)^2 + 6(x-2) = x^3 - 6x^2 + 12x - 8 - 5(x^2 - 4x + 4) + 6x - 12
= x^3 - 6x^2 + 12x - 8 - 5x^2 + 20x - 20 + 6x - 12
= x^3 - 11x^2 + 38x - 40
Since the expression is not equal to zero, x-2 is not a factor of x^3-5x^2+6x.

For option B, x+3:
(x+3)^3 - 5(x+3)^2 + 6(x+3) = x^3 + 9x^2 + 27x + 27 - 5(x^2 + 6x + 9) + 6x + 18
= x^3 + 9x^2 + 27x + 27 - 5x^2 - 30x - 45 + 6x + 18
= x^3 + 4x^2 - 36x
Since the expression is not equal to zero, x+3 is not a factor of x^3-5x^2+6x.

For option C, x+2:
(x+2)^3 - 5(x+2)^2 + 6(x+2) = x^3 + 6x^2 + 12x + 8 - 5(x^2 + 4x + 4) + 6x + 12
= x^3 + 6x^2 + 12x + 8 - 5x^2 - 20x - 20 + 6x + 12
= x^3 + x^2 - 2x
Since the expression is equal to zero, x+2 is a factor of x^3-5x^2+6x.

For option D, x^2+5x+6:
Plugging x^2+5x+6 into x^3-5x^2+6x gives us a long and complicated expression, which is not convenient to calculate. Hence, we cannot determine if x^2+5x+6 is a factor using direct substitution.

Therefore, the correct answer is C. x+2.