Exam-style Questions: Basic Algebra

1. It is given that:

f(x) = 8x3 + 18x2 + 3x -2.

a) Use the factor theorem to show that (x + 2) and (4x -1) are factors of f(x).

b) Given that the other factor of f(x) is (a + 1 ), find the roots of the equations.

(i) f(x) = 0,

(ii) f(a) = 0.

(Marks available: 5)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) Use the factor theorem

f (-2) = -64 + 72 - 6 - 2 = 0, thus (x+2) is a factor.

f (1/4) = 1/8 + 9/8 + 3/4 - 2 = 0, thus (4x-1) is a factor.

(2 marks)

b) (i) replacing the given equation with its factors:

f (x) = (x+2) (4x-1) (2x+1)

This gives roots of x = -2, x = 1/4, x = -1/2.

(1 mark)

(ii) change the above equation to a function of 2x

f (2x) = (2x+2) (8x-1) (4x+1)

This gives roots of x = -1, x = 1/8, x = -1/4.

(2 marks)

(Marks available: 5)