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?

Answer 1

To factor the expression \( x^3 - 5x^2 + 6x \), we can start by factoring 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 to \( -5 \) (the coefficient of \( x \)). These numbers are \( -2 \) and \( -3 \).

Thus, we can factor the quadratic as:

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

Now, substituting this back into the expression, we get:

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

Therefore, the complete factorization of \( x^3 - 5x^2 + 6x \) is:

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

Since \( x \) is given to be one of the factors, the remaining factors are \( (x - 2) \) and \( (x - 3) \).

Thus, another correct factor is either **\( (x - 2) \)** or **\( (x - 3) \)**.