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
$\text{f}(x) = 5x^3 + ax^2 - 31x + b$

where $a$ and $b$ are constants.

Given that $(5x + 2)$ is a factor of $\text{f}(x)$ ,
(a) show that $4a + 25b = -302$

(2 marks)

$\text{f}(x)$ leaves a remainder of $3$ when it is divided by $(x + 1)$
(b) Find the value of $a$ and the value of $b$ .

(4 marks)
(c) Using algebra, divide $\text{f}(x)$ by $(x - 3)$ , stating the quotient and the remainder.

(3 marks)
Level 4 · 9 marksTry it
2
The polynomial $\text{g}(x)$ is given by

$\text{g}(x) = 3x^3 - 5x^2 + 11x - 6$
(a) Use the factor theorem to show that $(3x - 2)$ is a factor of $\text{g}(x)$

(2 marks)
(b) Express $\text{g}(x)$ in the form

$\text{g}(x) = (3x - 2)(ax^2 + bx + c)$

where $a$ , $b$ and $c$ are integers to be found.

(2 marks)
(c) The integer $n$ is greater than $1$

Explain how your answer to part (b) shows that $3n^3 - 5n^2 + 11n - 6$ cannot be a prime number.

(2 marks)
Level 4 · 6 marksTry it
3
The polynomial $\text{f}(x)$ is defined by

$\text{f}(x) = 3x^3 + 7x^2 + 17x + 5$
(a) Use the factor theorem to show that $(3x + 1)$ is a factor of $\text{f}(x)$

(2 marks)
(b) Express $\text{f}(x)$ in the form

$\text{f}(x) = (3x + 1)(ax^2 + bx + c)$

where $a$ , $b$ and $c$ are constants to be found.

(2 marks)
(c) The integer $N$ is given by

$N = 3n^3 + 7n^2 + 17n + 5$

where $n$ is a positive integer.

Deduce from your answer to part (b) that $N$ is not a prime number.

(2 marks)
Level 4 · 6 marksTry it
4
(a) Show that $x = 3$ is a root of the equation $2x^{3} - x^{2} - 13x - 6 = 0$ . (1 mark)
(b) Hence solve the equation completely. (4 marks)
Level 4 · 5 marksTry it
5
In this question you must show detailed reasoning.

It is given that $f(x) = 2x^3 + x^2 - 25x + 12$ .
(a) Use the factor theorem to show that $(x + 4)$ is a factor of $f(x)$ .

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

(3 marks)
(c) Hence find the exact values of $y$ for which $2^{3y + 1} + 4^y - 25 \times 2^y + 12 = 0$ .

(4 marks)
Level 4 · 8 marksTry it

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