A-level · Algebra & functions

Polynomials and the factor theoremExam-style questions

Solving a cubic by finding one root first. Which whole numbers are worth trying, and why; the remainder theorem, which finds a remainder with one substitution; the factor theorem, and testing a factor like (2x − 1); dividing out a factor by matching coefficients, with a 0 for every missing power; finishing the cubic from the quadratic that is left; and reading a sketch from the factors, where a squared factor touches the axis.

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35 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The polynomial $f(x) = x^3 + ax^2 + bx + 30$ has $(x - 2)$ and $(x + 3)$ as factors.
(a) Find the values of $a$ and $b$ .
(b) Find the remaining linear factor.
Level 5 · 4 marksTry it
2
$f(x) = 3x^3 - 5x^2 - 26x - 8$ .
(a) Show that $(x + 2)$ is a factor of $f(x)$ . (2 marks)
(b) Factorise $f(x)$ completely. (3 marks)
(c) Hence solve $f(x) = 0$ . (2 marks)
Level 5 · 7 marksTry it
3
$f(x) = x^{3} - 3x^{2} - 10x + 24$ .
(a) Given that $f(2) = 0$ , factorise $f(x)$ completely. (4 marks)
(b) Hence solve the inequality $f(x) > 0$ . (2 marks)
Level 5 · 6 marksTry it
4
The polynomial $\mathrm{p}(x)$ is defined by

$\mathrm{p}(x) = 2x^{3} + 11x^{2} + 8x + k,$

where $k$ is a constant.
(a) Find the quotient when $\mathrm{p}(x)$ is divided by $(2x + 3)$ and show that the remainder is $(k + 6)$ .

(3 marks)
(b) It is given that $\displaystyle\int_{0}^{3} \frac{\mathrm{p}(x)}{2x + 3}\,\mathrm{d}x = a + \ln 243$ , where $a$ is an integer.

Find the values of $a$ and $k$ .

(6 marks)
Level 5 · 9 marksTry it
5
$f(x) = x^3 + ax^2 - 5x + b$ . Given that $(x - 1)$ is a factor of $f(x)$ , and that the remainder when $f(x)$ is divided by $(x + 2)$ is $-15$ , find the values of $a$ and $b$ . (6 marks)
Level 5 · 6 marksTry it

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