Expanding brackets
To expand means to multiply out brackets. Multiply every term inside by the term outside: a(b + c) = ab + ac. For two brackets, multiply each term in the first by each term in the second (FOIL): (x + p)(x + q) = x² + (p + q)x + pq.
Key idea
Index laws: xa × xb = xa+b, xa ÷ xb = xa−b, (xa)b = xab. Difference of two squares: a² − b² = (a + b)(a − b).
Factorising
To factorise is the reverse of expanding: write an expression as a product. First take out any common factor, then look for special forms such as the difference of two squares or a quadratic that factorises into two brackets.
Worked example
Expand and simplify (x + 5)(x − 2).
= x·x + x·(−2) + 5·x + 5·(−2)
= x² − 2x + 5x − 10
= x² + 3x − 10.
Now factorise x² + 3x − 10 back: two numbers with product −10 and sum +3 are +5 and −2, giving (x + 5)(x − 2). ✓
Simplifying
Collect like terms (same letters and powers) by adding coefficients. Simplify algebraic fractions by cancelling common factors on top and bottom.
Remember
- Multiply the sign as well as the number.
- x + x = 2x, but x × x = x².
- Always factorise fully — take out the biggest common factor first.