Algebra

Maths Pure 3 · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths Pure 3 topics

The idea

This topic collects four algebra tools. The modulus |x| is the size of x without its sign, so |x| is never negative. The graph of y = |ax + b| is a V shape: the part of the line below the x-axis is reflected above it.

Dividing polynomials gives a quotient and a remainder. The remainder and factor theorems give the remainder quickly, without doing the division.

Partial fractions split one algebraic fraction into simpler ones, with one term for each factor of the denominator. The binomial expansion of (1 + x)ⁿ for a negative or fractional n goes on for ever and is only valid when |x| < 1.

Rules to remember

If k > 0, |f(x)| = k means f(x) = k or f(x) = −k; and |x − a| < b means a − b < x < a + b
For |f(x)| = |g(x)| or |f(x)| > |g(x)|, square both sides to remove the modulus signs
Remainder theorem: when p(x) is divided by (x − a) the remainder is p(a); when divided by (ax − b) it is p(b/a)
Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0
Partial fractions: (x − a)(x − b) gives A/(x − a) + B/(x − b); a repeated factor (x − a)² gives A/(x − a) + B/(x − a)²; a factor (x² + c) gives (Bx + C)/(x² + c)
(1 + x)ⁿ = 1 + nx + [n(n − 1)/2!]x² + [n(n − 1)(n − 2)/3!]x³ + ..., valid for |x| < 1
(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ, valid for |bx/a| < 1

Common mistake

In (a + bx)ⁿ students take out a but forget to raise it to the power n, for example writing (4 + x)^½ as 4(1 + x/4)^½ instead of 2(1 + x/4)^½. Always write aⁿ outside the bracket first.

Worked example

Express (7x + 1)/[(x − 1)(x + 3)] in partial fractions.

Write (7x + 1)/[(x − 1)(x + 3)] ≡ A/(x − 1) + B/(x + 3), so 7x + 1 ≡ A(x + 3) + B(x − 1).
Put x = 1: 8 = 4A, so A = 2.
Put x = −3: −20 = −4B, so B = 5.
Check: 2(x + 3) + 5(x − 1) = 2x + 6 + 5x − 5 = 7x + 1.

Answer: 2/(x − 1) + 5/(x + 3)

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Chapter 1: Algebra

Keep going

Logarithmic and exponential functions →Take a Maths Pure 3 test