Question

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? x+3, x+2, x^2+5x+9, x-2

Answer 1

To find a factor of the polynomial \( x^3 - 5x^2 + 6x \), we can first factor out the common term:

\[
x^3 - 5x^2 + 6x = x(x^2 - 5x + 6)
\]

Next, we need to factor the quadratic \( x^2 - 5x + 6 \). We can look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of the linear term).

The factors of \( 6 \) that add to \( -5 \) are \( -2 \) and \( -3 \):

\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Now, substituting back, we have:

\[
x^3 - 5x^2 + 6x = x(x - 2)(x - 3)
\]

So, the complete factorization of the polynomial is:

\[
x(x - 2)(x - 3)
\]

Given that \( x \) itself is a factor, and we are looking for another linear factor among the options provided, we see that \( x - 2 \) is indeed a factor.

Among the choices provided:

- \( x + 3 \)
- \( x + 2 \)
- \( x^2 + 5x + 9 \)
- \( x - 2 \)

The correct answer is:

\[
\boxed{x - 2}
\]