Question

The polynomial function P(x) = x^3 - 5x^2 - 2x + 24 has a known factor of (x + 2). Write P(x) as a product of linear factors.

Answer 1

The polynomial function P(x) can be rewritten as the product of linear factors (x + 2)(x - 3)(x - 4) after dividing by the known factor (x + 2) and subsequently factoring the resulting quadratic equation. So, P(x) completely factored is (x + 2)(x - 3)(x - 4).

Step-by-step explanation:

To write the polynomial function P(x) = x^3 - 5x^2 - 2x + 24 as a product of linear factors, given that (x + 2) is a known factor, we will begin by performing polynomial division or using synthetic division to divide the polynomial by (x + 2).

Performing long division or synthetic division, we find:

1. Divide (x^3 - 5x^2 - 2x + 24) by (x + 2).
2. The result of this division is a quadratic polynomial (x^2 - 7x + 12).
3. Factor the quadratic polynomial to find the remaining linear factors. Factoring gives us (x - 3)(x - 4).
4. Thus, P(x) as a product of linear factors is (x + 2)(x - 3)(x - 4).

So, P(x) completely factored is (x + 2)(x - 3)(x - 4).