Question

Find the values of a and b, if (x – 1) and (x + 2) are both factors of \[(x^{3} + ax^{2} + bx - 6)\].

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 $$f(x) = x^3 + ax^2 + bx - 6$$.

Since $$(x - 1)$$ is a factor, $$f(1) = 0$$.

$$1 + a + b - 6 = 0$$

$$a + b = 5$$

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

$$-8 + 4a - 2b - 6 = 0$$

$$2a - b = 7$$

Solving the two equations gives $$a = 4$$ and $$b = 1$$.

The polynomial is $$x^3 + 4x^2 + x - 6 = (x - 1)(x + 2)(x + 3)$$.