Algebraic identities and factoring

Expanding and factoring: distributivity, the three algebraic identities, common factors and the completed-square form of a quadratic.

The outline, topic by topic

Each topic is weighted the way the exam weighs it.

Algebraic identity: what is it?
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An equality true for all values of the letters, used as a calculation shortcut: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².
It reads both ways: left to right to expand, right to left to factor. The classic mistake is to forget the middle term 2ab.
Expand (x − 4)².
- x² − 8x + 16
- x² − 16
- x² + 16
- x² − 8x − 16
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x² − 8x + 16
(x − 4)² = x² − 2 × 4x + 4² = x² − 8x + 16. x² − 16 forgets the middle term; the last term of a square is always positive.
(a + b)² = a² + … + b²
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(a + b)² = a² + 2ab + b²
(a + b)² = (a + b)(a + b) gives a² + ab + ba + b²: the middle term is the double product, never missing.
Expand: what is it?
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Turn a product into a sum by distributing each term of one factor over those of the other, then collecting like terms.
(x + 4)(x − 1) = x² − x + 4x − 4 = x² + 3x − 4. When you open a bracket preceded by a minus sign, every sign inside flips.
Expand (x + 3)(x − 3).
- x² + 9
- x² − 9
- x² − 6
- x² − 3
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x² − 9
This is (a + b)(a − b) = a² − b² with a = x and b = 3: x² − 9. The terms −3x and +3x cancel.