1

A student claims that $(x + 1)$ is a factor of $f(x) = x^{3} + 3x^{2} - 4$. Is the student correct? Justify your answer using the factor theorem. [2]

2

It is given that $f(x) = x^{3} - x^{2} - 8x - 4$ has $(x + 2)$ as a factor. Factorise $f(x)$ as far as possible, and hence write down the exact solutions of $f(x) = 0$, giving any non-integer solutions in surd form. [4]

3

It is given that $f(x) = x^{3} - 2x^{2} + kx + 5$ leaves a remainder of $11$ when divided by $(x - 3)$. Work out the value of $k$. [4]

4

Solve the equation $x^{3} - 7x + 6 = 0$. [5]

5

It is given that $f(x) = x^{3} + ax^{2} + bx - 10$, that $(x - 1)$ is a factor, and that $f(2) = 0$. Work out the values of $a$ and $b$. [5]