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:
\[ \textbf{x - 2} \]
1 answer