Final answer:
To find the value of k so that (x+2) is a factor of P(x), we use the remainder theorem to discover that k equals 6. Then, by performing polynomial division with k=6, we find the other two factors of P(x) to be (x-6) and (x-2).
Step-by-step explanation:
To find the value of k for which (x+2) is a factor of the polynomial P(x) = x³ - kx² + 4x + 24, we can use the remainder theorem. According to the remainder theorem, if (x+2) is a factor of P(x), then P(-2) = 0. Substituting -2 into P(x) gives us:
P(-2) = (-2)³ - k(-2)² + 4(-2) + 24 = 0
After simplifying, we get:
-8 - 4k + 8 + 24 = 0
-4k + 24 = 0
k = 6
Next, to find the other factors of P(x) with k=6, we now have P(x) = x³ - 6x² + 4x + 24. We can perform polynomial long division or synthetic division with the divisor x+2 to obtain the quotient polynomial, which will give us the other factors.
After division, the quotient polynomial is x² - 8x + 12, which can be factored further into (x-6)(x-2). Therefore, the other two factors of P(x) are (x-6) and (x-2).