Question

Use factor theorem to show that (x + 2) and (2x – 3) are factors of \[(2x^{2} + x - 6)\].

Answer 1

Given: $$f(x) = 2x^2 + x - 6$$.

To prove: $$(x + 2)$$ and $$(2x - 3)$$ are factors of $$f(x)$$.

Proof:

1. By the Factor Theorem: $$(x - \alpha)$$ is a factor of $$f(x) \Leftrightarrow f(\alpha) = 0$$.
2. $$f(-2) = 2(-2)^2 + (-2) - 6 = 4 - 2 - 6 = 0$$.
3. Therefore, $$(x + 2)$$ is a factor of $$f(x)$$.
4. By the Factor Theorem: $$(ax + b)$$ is a factor of $$f(x) \Leftrightarrow f\left(\frac{-b}{a}\right) = 0$$.
5. $$f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^2 + \frac{3}{2} - 6 = \frac{9}{2} + \frac{3}{2} - 6 = 0$$.
6. Therefore, $$(2x - 3)$$ is a factor of $$f(x)$$.
7. Hence, $$f(x) = (x + 2)(2x - 3)$$.

Hence proved.