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) Responses x+2 x plus 2 x2+5x+6 x squared plus 5 x plus 6 x+3 x plus 3 x−2

Answer 1

To find a linear factor of the polynomial \( x^3 - 5x^2 + 6x \), we first factor the expression completely.

We can start by factoring out the greatest common factor (GCF), which is \( x \):

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

Now, we need to factor the quadratic \( x^2 - 5x + 6 \). We look for two numbers that multiply to 6 and add to -5. The numbers -2 and -3 meet this criterion.

Therefore, we can factor the quadratic as:

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

Putting this together, we have:

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

Now, the complete factorization of the polynomial is:

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

The linear factors are \( x \), \( x - 2 \), and \( x - 3 \).

Given that \( x \) is one of the linear factors, the other correct factors are \( x - 2 \) and \( x - 3 \). Among the provided choices, the factor:

- \( x - 2 \)

is indeed one of the factors of the polynomial \( x^3 - 5x^2 + 6x \).

So the correct response is:

**x - 2**