Question

Using factor theorem, show that (x – 3) is a factor of \[(x^{3} + x^{2} - 17x + 15)\].

Answer 1

Given: \[f(x) = (x^{3} + x^{2} - 17x + 15)\]

To prove: (x – 3) 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 let $$f_1(x) = x^3 + x^2 - 17x + 15$$.

$$f_1(3) = 27 + 9 - 51 + 15 = 0$$.

Therefore, $$(x - 3)$$ is a factor of $$f_1(x)$$.

Hence proved.