Question

Using factor theorem, show that (x + 1) is a factor of \[(x^{3} + 4x^{2} + 5x + 2)\].

Answer 1

Given: \[f(x) = (x^{3} + 4x^{2} + 5x + 2)\]

To prove: (x + 1) 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 $$f_2(x) = x^3 + 4x^2 + 5x + 2$$.

$$f_2(-1) = -1 + 4 - 5 + 2 = 0$$.

Therefore, $$(x + 1)$$ is a factor of $$f_2(x)$$.

Hence proved.