To find a factor of the polynomial \( x^3 - 5x^2 + 6x \), we can first factor out the common term \( x \):
\[ x^3 - 5x^2 + 6x = x(x^2 - 5x + 6) \]
Now, we need to factor the quadratic \( x^2 - 5x + 6 \). We are looking for two numbers that multiply to \( 6 \) and add to \( -5 \). These numbers are \( -2 \) and \( -3 \), so we can factor the quadratic as follows:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]
Therefore, the complete factorization of the original polynomial is:
\[ x^3 - 5x^2 + 6x = x(x - 2)(x - 3) \]
Now we know the factors are \( x \), \( x - 2 \), and \( x - 3 \). Given these factors, the question asks for another correct factor.
If \( x \) is one of the linear factors, then from the options provided, the other possible linear factors are:
- \( x - 2 \)
- \( x - 3 \)
The response options provided were:
- \( x + 3 \)
- \( x^2 + 5x + 6 \)
- \( x - 2 \)
- \( x + 2 \)
Among these options, the only correct factor of \( x^3 - 5x^2 + 6x \) that corresponds with our analysis is:
\( x - 2 \).