# Exam-style Questions: Basic Algebra

**1. It is given that:**

f(x) = 8x^{3} + 18x^{2} + 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 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)**