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)

x−2

x+3

x+2

x^2+5x+6

1 answer

The factor theorem states that if a polynomial f(x) has a factor (x-a), then f(a) = 0.

So, plugging in x=-2 into the given polynomial,
(-2)^3 - 5(-2)^2 + 6(-2) = -8 - 20 - 12 = -40.

Therefore, x + 2 is NOT a factor of the given polynomial.

Similarly, plugging in x = -3 into the given polynomial,
(-3)^3 - 5(-3)^2 + 6(-3) = -27 - 45 - 18 = -90.

Therefore, x + 3 is NOT a factor of the given polynomial.

Now, plugging in x=2 into the given polynomial,
(2)^3 - 5(2)^2 + 6(2) = 8 - 20 + 12 = 0.

Therefore, x - 2 is a factor of the given polynomial.

Finally, (x^3-5x^2+6x) / (x-2) = x^2 - 3x.

So, x^2 - 3x is another correct factor.