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# A-Level Maths | Cubics
- URL: https://www.a-level-maths-tutor.com/a-level-maths-cubics/
- Published: 2026-04-28T12:00:19.000Z
- Updated: 2026-08-15T10:17:52.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

Question 1

Divide the cubic by the linear factor, and fully factorise the cubic if possible.

(a) $x^3+2x^2-x-2$ by $x+1$

(b) $x^3+2x^2-9x+2$ by $x-2$

(c) $20+x+3x^2+x^3$ by $x+4$

(d) $2x^3-x^2-4x+3$ by $x-1$

(e) $6x^3-19x^2-73x+90$ by $x-5$

(f) $-x^3+5x^2+10x-8$ by $x+2$

(g) $x^3-2x+21$ by $x+3$

(h) $3x^3+16x^2+72$ by $x+6$

Show Answers 

(a) Quotient $x^2+x-2$; fully factorised: $(x-1)(x+1)(x+2)$

(b) Quotient $x^2+4x-1$ (does not factorise further over rationals)

(c) Quotient $x^2-x+5$ (does not factorise further — no real roots)

(d) Quotient $2x^2+x-3$; fully factorised: $(x-1)^2(2x+3)$

(e) Quotient $6x^2+11x-18$ (does not factorise further over rationals)

(f) Quotient $-x^2+7x-4$ (does not factorise further over rationals)

(g) Quotient $x^2-3x+7$ (does not factorise further — no real roots)

(h) Quotient $3x^2-2x+12$ (does not factorise further — no real roots)

Question 2

Use the factor theorem to determine whether or not

(a) $(x-1)$ is a factor of $x^3+2x^2-2x-1$

(b) $(x+2)$ is a factor of $x^3-5x^2-9x+2$

(c) $(x-3)$ is a factor of $x^3-x^2-14x+27$

(d) $(x+6)$ is a factor of $2x^3+13x^2+2x-24$

(e) $(2x+1)$ is a factor of $2x^3-5x^2+7x-14$

(f) $(3x-2)$ is a factor of $2-17x+25x^2-6x^3$

Show Answers 

(a) $f(1)=0$ — yes, a factor

(b) $f(-2)=-8$ — no, not a factor

(c) $f(3)=3$ — no, not a factor

(d) $f(-6)=0$ — yes, a factor

(e) $f(-\\frac{1}{2})=-19$ — no, not a factor

(f) $f(\\frac{2}{3})=0$ — yes, a factor

Question 3

Use the remainder theorem to find the remainder obtained in dividing

(a) $x^3+4x^2-x+6$ by $x-2$

(b) $x^3-2x^2+7x+1$ by $x+1$

(c) $2x^3+x^2-9x+17$ by $x+5$

(d) $8x^3+4x^2-6x-3$ by $2x-1$

(e) $2x^3-3x^2-20x-7$ by $2x+1$

(f) $3x^3-6x^2+2x-7$ by $3x-2$

Show Answers 

(a) $28$

(b) $-9$

(c) $-163$

(d) $-4$

(e) $2$

(f) $-\\frac{67}{9}$

Question 4

$$f(x) \\equiv x^3-2x^2-11x+12$$

(a) Show that $(x-1)$ is a factor of $f(x)$.

(b) Hence, express $f(x)$ as the product of three linear factors.

Show Answers 

(a) $f(1) = 1-2-11+12 = 0$, so $(x-1)$ is a factor.

(b) Dividing $f(x)$ by $(x-1)$ gives $x^2-x-12$, which factorises as $(x-4)(x+3)$.  
$$f(x) = (x-1)(x-4)(x+3)$$

Question 5

Given that $x=-2$ is a solution to the equation $$g(x)\\equiv x^3+7x^2+7x-6=0$$

(a) express $g(x)$ as the product of a linear factor and a quadratic factor,

(b) find, to 2 decimal places, the other two solutions to the equation $g(x)=0$.

Show Answers 

(a) Dividing $g(x)$ by $(x+2)$ gives $x^2+5x-3$.  
$$g(x) = (x+2)(x^2+5x-3)$$

(b) Solving $x^2+5x-3=0$ by the quadratic formula:  
$$x = \\frac{-5\\pm\\sqrt{25+12}}{2} = \\frac{-5\\pm\\sqrt{37}}{2}$$ $$x \\approx 0.54 \\text{ or } x \\approx -5.54$$

Question 6

By first finding a linear factor, fully factorise $$x^3-2x^2-5x+6$$

Show Answers 

Testing $x=1$: $1-2-5+6=0$, so $(x-1)$ is a factor.

Dividing by $(x-1)$ gives $x^2-x-6 = (x-3)(x+2)$.

$$x^3-2x^2-5x+6 = (x-1)(x-3)(x+2)$$

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