Question

If $$(x - 2)$$ is a factor of $$2x^3 - x^2 - px - 2$$, 

1. find the value of $$p$$ 
2. with the value of $$p$$, factorise the above expression completely.

Answer 1

Let $$f(x) = 2x^3 - x^2 - px - 2.$$

Since $$(x - 2)$$ is a factor, the Factor Theorem gives: $$f(2) = 0.$$

Substituting $$x=2$$: $$f(2) = 2(2)^3 - (2)^2 - 2p - 2$$

$$= 16 - 4 - 2p - 2$$

$$= 10 - 2p$$

Thus, $$10 - 2p = 0 \Rightarrow p = 5.$$

Substituting $$p = 5$$: $$2x^3 - x^2 - 5x - 2 = (x - 2)(2x^2 + 3x + 1).$$

Since $$2x^2 + 3x + 1 = (2x + 1)(x + 1),$$ the complete factorisation is $$2x^3 - x^2 - 5x - 2 = (x - 2)(2x + 1)(x + 1).$$