Question

Renee is asked to find a factor of x3−5x2+6x, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)%0D%0AResponses%0D%0A%0D%0Ax+3%0D%0Ax plus 3%0D%0A%0D%0Ax+2%0D%0Ax plus 2%0D%0A%0D%0Ax−2%0D%0Ax minus 2%0D%0A%0D%0Ax2+5x+6

Answer 1

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

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

Next, we need to factor the quadratic \( x^2 - 5x + 6 \). We look for two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of the linear term). The numbers -2 and -3 satisfy these conditions:

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

Putting it all together, we have:

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

This means the factors of the original polynomial are \( x \), \( x - 2 \), and \( x - 3 \).

Given that \( x \) is one of the linear factors, the other factors of the polynomial are \( x - 2 \) and \( x - 3 \).

Now let's look at the response options given:

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

From this list, the correct factor from our factoring process is:

\[
\textbf{x - 2}
\]