Question

If \[2x^{3} - 3x^{2} - 3x + 2 = (2x - 1)(x^{2} + ax + b):\] 

1. using Remainder and Factor theorem, find the value of a and b.
2. hence, factorise the polynomial \[2x^{3} - 3x^{2} - 3x + 2\] completely.

Answer 1

Given: $$2x^3 - 3x^2 - 3x + 2 = (2x - 1)(x^2 + ax + b)$$.

To find: The values of $$a$$ and $$b$$ and the complete factorisation.

Proof:

1. By the Remainder Theorem: When a polynomial $$f(x)$$ is divided by $$(x - \alpha)$$, then the remainder is $$f(\alpha)$$.
2. Let $$f(x) = 2x^3 - 3x^2 - 3x + 2$$.
3. $$f\left(\frac{1}{2}\right) = \frac{1}{4} - \frac{3}{4} - \frac{3}{2} + 2 = 0$$.
4. Therefore, $$(2x - 1)$$ is a factor of $$f(x)$$.
5. On dividing $$f(x)$$ by $$(2x - 1)$$, the quotient is $$x^2 - x - 2$$.
6. Comparing $$x^2 + ax + b$$ with $$x^2 - x - 2$$, we obtain $$a = -1$$ and $$b = -2$$.
7. $$x^2 - x - 2 = (x - 2)(x + 1)$$.
8. Hence, $$2x^3 - 3x^2 - 3x + 2 = (2x - 1)(x - 2)(x + 1)$$.

Hence proved.