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 $\text{P}(x)$ is given by

$\text{P}(x) = 6x^3 - x^2 - 14x + 8$

Use the factor theorem to show that $(3x - 2)$ is a factor of $\text{P}(x)$

(2 marks)
Level 2 · 2 marksTry it
2
$\text{f}(x) = (2x - 1)(x^2 + 6x + k) - 18 \qquad \text{where } k \text{ is a constant}$

The cubic $\text{f}(x)$ has $(x + 4)$ as a factor.

Find the value of $k$ .

(3 marks)
Level 2 · 3 marksTry it
3
The cubic polynomial $f(x)$ is defined by $f(x) = 2x^3 - x^2 - 13x - 6$ .
(a) Use the factor theorem to show that $(2x + 1)$ is a factor of $f(x)$ .

(1 mark)
(b) Express $f(x)$ in fully factorised form.

(3 marks)
Level 2 · 4 marksTry it
4
$\text{f}(x) = (3x + 5)(x^2 - 6x + k) - 20 \qquad \text{where } k \text{ is a constant}$

Given that $(x + 1)$ is a factor of $\text{f}(x)$ , find the value of $k$ .

(3 marks)
Level 2 · 3 marksTry it
5
(a) Show that $2x^3 - 3x^2 - 11x + 6$ can be written as $(x - 3)\left(2x^2 + 3x - 2\right)$ . (3 marks)
(b) Hence solve the equation $2x^3 - 3x^2 - 11x + 6 = 0$ . (2 marks)
Level 3 · 5 marksTry it

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