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Algebra fluency: expanding and factorising
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Further Pure moves fast and expects your algebra to be automatic. Expanding and factorising come up in almost every question, so this is where to start.
Expanding brackets
Multiply every term in one bracket by every term in the other.
(2x + 3)(x − 5) = 2x2 − 10x + 3x − 15 = 2x2 − 7x − 15
Squares: (x − 4)2 = (x − 4)(x − 4) = x2 − 8x + 16, not x2 + 16.
Three brackets: expand two first, then multiply by the third.
(2x + 3)(x − 5) = 2x2 − 10x + 3x − 15 = 2x2 − 7x − 15
Squares: (x − 4)2 = (x − 4)(x − 4) = x2 − 8x + 16, not x2 + 16.
Three brackets: expand two first, then multiply by the third.
Factorising
Common factor: take out the highest common factor. 6x2 + 9x = 3x(2x + 3)
Quadratic, x2 + bx + c: find two numbers that multiply to c and add to b.
x2 + x − 12: 4 × (−3) = −12 and 4 + (−3) = 1, so (x + 4)(x − 3).
Quadratic, x2 + bx + c: find two numbers that multiply to c and add to b.
x2 + x − 12: 4 × (−3) = −12 and 4 + (−3) = 1, so (x + 4)(x − 3).
Special cases
Difference of two squares: a2 − b2 = (a + b)(a − b)
9x2 − 25 = (3x + 5)(3x − 5)
Always look for a common factor first: 2x2 − 8 = 2(x2 − 4) = 2(x + 2)(x − 2)
ax2 + bx + c: find two numbers that multiply to ac and add to b, then split the middle term.
9x2 − 25 = (3x + 5)(3x − 5)
Always look for a common factor first: 2x2 − 8 = 2(x2 − 4) = 2(x + 2)(x − 2)
ax2 + bx + c: find two numbers that multiply to ac and add to b, then split the middle term.
Factorise 3x2 − 10x − 8
- ac = 3 × (−8) = −24. Find numbers that multiply to −24 and add to −10: −12 and 2.
- Split: 3x2 − 12x + 2x − 8
- Group: 3x(x − 4) + 2(x − 4)
- = (3x + 2)(x − 4)
Answer: (3x + 2)(x − 4)
Multiply every term by every term when expanding. When factorising, take out common factors first, then look for a quadratic or a difference of two squares.
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