IP EMaths Notes (Upper Sec, Year 3-4): 01) Algebraic Tools

Keep algebra tidy so every later topic - graphs, trigonometry, variation - stays manageable. These identities should come out automatically.

Key skills to lock in

• Apply the distributive law: \(a(b + c) = ab + ac\).
• Recognise perfect-square and difference-of-squares patterns, e.g. \(x^{2} - 9 = (x - 3)(x + 3)\).
• Factor by grouping to reveal hidden common factors.
• Simplify expressions with multiple parentheses before substituting values.

Worked example - Rewrite for evaluation

Rewrite \(3(2x - 5) + 4(x + 1)\) in simplified form, then find its value when \(x = -2\).

1. Expand each product: \(3(2x - 5) = 6x - 15\) and \(4(x + 1) = 4x + 4\).
2. Combine like terms: \((6x - 15) + (4x + 4) = 10x - 11\).
3. Substitute \(x = -2\): \(10(-2) - 11 = -20 - 11 = -31\).

So the simplified expression is \(10x - 11\), and its value at \(x = -2\) is \(-31\).

Try this

Factorise \(2x^{2} - 5x - 12\) completely and state the values of \(x\) that make the expression zero.