Summary

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Square of a sum and a difference, difference of squares, and cube of a sum and a difference — very useful formulas that let you quickly expand or factor expressions with powers.

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  • Term1
  • Statement5
  • Important2
  • Problem28
  • Updated
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Summary
Practice
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Squares — memorize these
Square of a sum and a difference(a±b)2=a2±2ab+b2(a \pm b)^2 = a^2 \pm 2ab + b^2Examples
Difference of squaresa2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)Examples
Cubes — do not memorize, but learn to recognize
Cube of a sum and a difference(a±b)3=a3±3a2b+3ab2±b3(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3Examples

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Formulas that let you quickly “unpack” compact powered expressions into expansions or, going the other way, “pack” long sums into a compact form. These formulas save you from doing routine calculations by hand.
  • Plus-minus sign
    Do not memorize 4 separate formulas: the square of a sum, the square of a difference, the cube of a sum, and the cube of a difference. It is enough to remember 2 formulas if you use the plus-minus sign, because nothing changes except the signs:
    (a±b)2=a2±2ab+b2(a±b)3=a3±3a2b+3ab2±b3(a \pm b)^2 = a^2 \pm 2ab + b^2 \\ (a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3
    In sums, all signs are always pluses. In a difference, the minus sign always comes right after the first term in the expansion. In the cube case, it also appears in front of the last term.
  • The difference of squares stands apart
    Formulas whose names start with the degree, like square and cube, have a similar shape, and they can be derived naturally by expanding brackets (a+b)2=(a+b)(a+b)=…(a+b)^2 = (a+b)(a+b) = \ldots
    But the difference of squares a2−b2a^2-b^2 stands apart. First, it factors into brackets with a plus and a minus. Second, from the form a2−b2a^2 - b^2 you cannot naturally and explicitly get the product (a + b)(a – b) from the difference itself.
  • The degree matches the coefficient
    In the square or cube of a sum/difference, the degree (second or third) also appears as a coefficient in the expansion. For the square it is 2, and for the cube it is 3:
    (a±b)2=a2±2ab+b2(a±b)3=a3±3a2b+3ab2±b3(a \pm b)^{\normalsize\brand{2}} = a^2 \pm \brand{2}ab + b^2 \\ (a \pm b)^{\normalsize\brand{3}} = a^3 \pm \brand{3}a^2b + \brand{3}ab^2 \pm b^3
    The coefficient is also easy to remember from the geometric derivation. For the square formulas, we build a square, and in the process two rectangles appear. For the cube formulas, we build a cube, and in the process two kinds of three parallelepipeds appear.
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