Given that \( x - 3 \) is a factor of the polynomial \( x^3 - 5x^2 - 2x + 24 \), we can use polynomial long division to divide \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \).
Performing the division, we would find the quotient polynomial. Then, we would look for the remaining factors of that quotient polynomial. However, we can also apply the factor theorem and synthetic division to check for all other potential linear factors.
Since we are provided with the possible options, we can substitute the values of \( x \) based on the proposed factors to see if any of these will yield a remainder of \( 0 \).
Here are the options and their evaluations:
-
For \( x - 2 \): \[ P(2) = 2^3 - 5(2^2) - 2(2) + 24 = 8 - 20 - 4 + 24 = 8 \quad (\text{not a factor}) \]
-
For \( x + 2 \): \[ P(-2) = (-2)^3 - 5(-2)^2 - 2(-2) + 24 = -8 - 20 + 4 + 24 = 0 \quad (\text{is a factor}) \]
-
For \( x + 4 \): \[ P(-4) = (-4)^3 - 5(-4)^2 - 2(-4) + 24 = -64 - 80 + 8 + 24 = -112 \quad (\text{not a factor}) \]
-
For \( x^2 - 8x + 22 \): This polynomial is not linear, and we won't evaluate it directly. However, we can check if it is related by the degree of already known factors.
Since \( x + 2 \) yielded \( 0 \), it confirms that \( x + 2 \) is indeed a factor. Therefore, the correct response is:
x + 2
0 answers