Question

Using factor theorem, show that (3x – 2) is a factor of \[(3x^{3} + x^{2} - 20x + 12)\].

Answer 1

Given: \[f(x) = (3x^{3} + x^{2} - 20x + 12)\]

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

Proof:

By the Factor Theorem: Let $$f(x)$$ be a given polynomial and $$\alpha$$ be a given real number.

Then, $$(x - \alpha)$$ is a factor of $$f(x) \Leftrightarrow f(\alpha) = 0$$.

For the zero of $$3x - 2$$ is $$x = \frac{2}{3}$$.

Let $$f_3(x) = 3x^3 + x^2 - 20x + 12$$.

$$f_3\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^3 + \left(\frac{2}{3}\right)^2 - 20\left(\frac{2}{3}\right) + 12 = 0$$.

Therefore, $$(3x - 2)$$ is a factor of $$f_3(x)$$.

Hence proved.