Question

Renee is asked to find a factor of x3−5x2+6x
𝑥
3
−
5
𝑥
2
+
6
𝑥
, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)
Responses

x−2
𝑥
−
2
x minus 2

x+2
𝑥
+
2
x plus 2

x2+5x+6
𝑥
2
+
5
𝑥
+
6
x squared plus 5 x plus 6

x+3

Answer 1

To find another factor of the polynomial \( x^3 - 5x^2 + 6x \), we first need to determine its roots and then factor it.

We can start by factoring out the greatest common factor from the polynomial:

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

Now, 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 to \( -5 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( -2 \) and \( -3 \). Therefore, we can factor the quadratic as follows:

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

Combining this with the factored out \( x \), we have:

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

Thus, the complete factorization is:

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

From the factors we found, we know that \( x \) is one factor and \( x - 2 \) and \( x - 3 \) are the other factors.

Since the question asks for another correct factor given that \( x \) is one of the linear factors, the correct response from the options provided is:

**\( x - 2 \)**.