# Regarding the factor and remainder theorem?

## The polynomial $4 {x}^{3} - 4 {x}^{2} + 3 x + a$, where a is a constant, is denoted by p(x). It is given that p(x) is divisible by $2 {x}^{2} - 3 x + 3$. (i) Find the value of a. (I'm done with this part of the question. The value of a is 3. It's the second part that's troubling me.) (ii) When a has this value, solve the inequality p(x) < 0, justifying your answer.

Aug 24, 2016

$x < - \frac{1}{2}$

#### Explanation:

We know that the function can be written $\left(2 {x}^{2} - 3 x + 3\right) \left(2 x + 1\right)$
What does the graph look like?
The determinant of the quadratic part indicates no real solution so the graph only crosses the X axis at $x = - \frac{1}{2}$
When x= 0 the function is positive
Therefore it is negative if $x < - \frac{1}{2}$
Hope this helps