A-Level Maths | Cubics

Question 1

Polynomial long division — divide the cubic by the linear factor to find the quotient

Find the quotient obtained in dividing

(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 $(x + 6)$

Question 2

Factor theorem — if $(x - a)$ is a factor, then $f(a) = 0$

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 $(2 - 17x + 25x^2 - 6x^3)$

Question 3

Remainder theorem — if $(x - a)$ is not a factor, then $f(a)$ equals the remainder when dividing

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 $(3x - 2)$

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.

Question 5

$g(x) \equiv x^3 + 7x^2 + 7x - 6$

Given that $x = -2$ is a solution to the equation $g(x) = 0$,

(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 $g(x) = 0$.

Question 6

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