Question

Using factor theorem, show that (3 – 2x) is a factor of \[(2x^{3} - 9x^{2} + x + 12)\].

Answer 1

Given: \[f(x) = (2x^{3} - 9x^{2} + x + 12)\]

To prove: (3 – 2x) is a factor and the complete factorisation of f(x)

Proof:

By Factor Theorem: $$(ax + b)$$ is a factor of $$f(x) \Leftrightarrow f\left(\frac{-b}{a}\right) = 0$$

For $$(3 - 2x)$$, $$a = -2$$ and $$b = 3$$.

$$-\frac{b}{a} = -\frac{3}{-2} = \frac{3}{2}$$

$$f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 - 9\left(\frac{3}{2}\right)^2 + \frac{3}{2} + 12$$

$$= 2 \times \frac{27}{8} - 9 \times \frac{9}{4} + \frac{3}{2} + 12$$

$$= \frac{27}{4} - \frac{81}{4} + \frac{6}{4} + \frac{48}{4}$$

$$= \frac{27 - 81 + 6 + 48}{4} = 0$$

Therefore, $$(3 - 2x)$$ is a factor of $$f(x)$$ [By Factor Theorem].

Hence proved.