Understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms and factors.

Recognise the difference between an equation and an identity, and show algebraic expressions are equivalent.

e.g. show that \((x+1)^2 + 2 = x^2 + 2x + 3\)

Use algebra to construct arguments.

Use algebra to construct proofs and arguments.

e.g. prove that the sum of three consecutive integers is a multiple of 3.

Simplify algebraic expressions by collecting like terms.

e.g. \(2a + 3a = 5a\)

Simplify algebraic products and quotients.

e.g. \(a×a×a=a^3\)

\(2a×3a=6ab\)

\(a^2×a^3=a^5\)

\(3a^3÷a=3a^2\)

[see also Laws of indices, 3.01c]

Simplify algebraic products and quotients using the laws of indices.

e.g. \(a^{\frac{1}{2}} × 2a^{-3} = 2a^{\frac{-5}{2}}\)

\(2a^2b^3 ÷ 4a^{-3}b = \dfrac{1}{2}a^5b^2\)

Simplify algebraic expressions by muliplying a single term over a bracket.

e.g. \(2(a+3b) = 2a+6b\)

\(2(a+3b) + 3(a-2b) = 5a\)

Expand products of two binomials.

e.g. \((x-1)(x-2) = x^2 - 3x + 2\)

\(a+2b)(a-b) = a^2 + ab + 2b^2\)

Expand products of more than two binomials.

e.g. \((x+1)(x-1)(2x+1) = 2x^3 + x^2 - 2x - 1\)

Take out common factors.

e.g. \(3a - 9b = 3(a-3b)\)

\(2x + 3x^2 = x(2+3x)\)

Factorise quadratic expressions of the form \(x^2 + bx + c\).

e.g. \(x^2 - x - 6 = (x-3)(x+2)\)

\(x^2 - 16 = (x-4)(x+4)\)

\(x^2 - 3 = (x-\sqrt{3})(x+\sqrt{3})\)

Factorise quadratic expressions of the form \(ax^2 + bx + c\) (where \(a \neq 0\) or \(1\)).

e.g. \(2x^2 + 3x - 2 = (2x-1)(x+2)\)

Complete the square on a quadratic expression.

e.g. \(x^2 + 4x - 6 = (x+2)^2 - 10\)

\(2x^2 + 5x + 1 = 2\Big(x+\dfrac{5}{4}\Big)^2 - \dfrac{17}{8}\)

Simplify and manipulate algebraic fractions.

e.g. Write \(\dfrac{1}{n-1} + \dfrac{n}{n+1}\) as a single fraction.

Simplify \(\dfrac{n^2+2n}{n^2+n-2}\)