Algebraic expressions

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

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The idea

Algebra uses letters for numbers, and the same rules of arithmetic apply. A term is one part of an expression, such as 5x². An expression has no equals sign; an equation is true for some values; an identity (≡) is true for every value.

Two skills are opposites. Expanding removes brackets by multiplying every term inside by what is outside. Factorising puts brackets back by taking out what the terms share. You can always check a factorisation by expanding it again.

With algebraic fractions, treat them like number fractions: factorise first, cancel only whole factors, and use a common denominator to add or subtract.

Rules to remember

Like terms have exactly the same letters and powers: 3x² and 5x² are like terms, 3x² and 3x are not.
When substituting a negative number, put it in brackets: if a = −3 then a² = (−3)² = 9.
Two brackets: (a + b)(c + d) = ac + ad + bc + bd. For three brackets, expand two first, then multiply by the third.
x² + bx + c: find two numbers that multiply to c and add to b.
ax² + bx + c: find two numbers that multiply to a × c and add to b, split the middle term, then factorise in pairs.
Difference of two squares: a² − b² = (a + b)(a − b).
A minus sign in front of a bracket changes every sign inside: −2(x − 4) = −2x + 8.

Common mistake

Cancelling single terms in a fraction, such as crossing out the x² in (x² − 9)/(x² + x − 6). You can only cancel a whole bracket that multiplies the top and the bottom, so factorise first.

Worked example

Simplify fully (2x² + 7x + 3)/(x² − 9).

Top: 2 × 3 = 6, and 6 and 1 multiply to 6 and add to 7. 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Bottom: difference of two squares, x² − 9 = (x + 3)(x − 3).
Cancel the common factor (x + 3) from top and bottom.

Answer: (2x + 1)/(x − 3)

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