Question

If (x + 2) and (x + 3) are factors of \[x^{3} + ax + b\], find the values of a and b.

Answer 1

Given: The two polynomials.

To find: The values of $$a$$ and $$b$$ in each case.

By the Factor Theorem: $$(x - \alpha)$$ is a factor of $$f(x) \Leftrightarrow f(\alpha) = 0$$.

For let $$g(x) = x^3 + ax + b$$.

Since $$(x + 2)$$ is a factor, $$g(-2) = 0$$.

$$-8 - 2a + b = 0$$

$$b - 2a = 8$$

Since $$(x + 3)$$ is a factor, $$g(-3) = 0$$.

$$-27 - 3a + b = 0$$

$$b - 3a = 27$$

Solving the two equations gives $$a = -19$$ and $$b = -30$$.

The polynomial is $$x^3 - 19x - 30 = (x + 2)(x + 3)(x - 5)$$.