Polynomials & Factor Theorem (AQA GCSE Further Maths): Exam Questions

$\mathrm{f}\left(x\right)=3x^3–2x^2–7x–2$

Use the factor theorem to show that $(3x+1)$ is a factor of $\mathrm{f}\left(x\right)$.

Factorise $\mathrm{f}\left(x\right)$ fully.

$\mathrm{f}\left(x\right)=200x^3+100x^2-18x-9$

Use the factor theorem to show that $\left(2x+1\right)$ is a factor of $\mathrm{f}\left(x\right)$.

Hence solve  $\mathrm{f}\left(x\right)=0$.

$\left(x-3\right)$ is a factor of $x^3-8x^2+ax+42$  where $a$ is an integer.

Show that the value of $a$ is 1.

Hence, factorise fully   $x^3-8x^2+x+42$.

Solve $x^3+x^2-26x+24=0$

You must show your working.

The curve $y=2x^3-x^2-32x+16$ has an $x$-intercept of $\frac{1}{2}$.

Use this information to find the other two $x$-intercepts.

$\mathrm{f}\left(x\right)=x^3+p{x}^2+q$ where $p$ and $q$ are integers.

The equation $\mathrm{f}\left(x\right)=0$ has the solution $x=2$ and a different non-zero solution that is a repeated solution.

By factorising $\mathrm{f}\left(x\right)$, or otherwise, find the values of $p$ and $q$.

By first finding a factor of $x^3-x^2+3x-10$, show that the equation $x^3-x^2+3x-10=0$ has one solution only.

$\mathrm{f}\left(x\right)=a{x}^3-b{x}^2-a^{3}x+a^{2}b$ where $a$ and $b$ are non-zero integers.

Work out $\mathrm{f}\left(\frac{b}{a}\right)$.

Factorise $\mathrm{f}\left(x\right)$ fully, giving your answer in terms of $a$ and $b$.