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^2+5x+6, x+3, x+2, or x-2

Answer 1

To find a factor of the polynomial \( x^3 - 5x^2 + 6x \), we can start by factoring it.

First, we can factor out the greatest common factor, which is \( x \):

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

Next, we need to factor the quadratic \( x^2 - 5x + 6 \). We are looking for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( x \)). The numbers that meet these criteria are \( -2 \) and \( -3 \). Thus, we can factor the quadratic as follows:

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

Putting it all together, we have:

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

Now that we fully factored the polynomial, we find that the factors of \( x^3 - 5x^2 + 6x \) are \( x \), \( x - 2 \), and \( x - 3 \).

Now, looking at the options provided:
1. \( x^2 + 5x + 6 \)
2. \( x + 3 \)
3. \( x + 2 \)
4. \( x - 2 \)

The correct factors that we found are \( x \), \( x - 2 \), and \( x - 3 \). Therefore, \( x - 2 \) is indeed one of the correct factors.

Thus, the answer to the question of which of the listed options is another correct factor is:

\[
\boxed{x - 2}
\]