Sums & differences of cubes

Japanese school year: Math II

What you learn

Learn how to factor cubic expressions into linear and quadratic factors using the sum and difference of cubes formulas. This technique is widely utilized when solving higher-degree equations and simplifying algebraic fractions. Prior knowledge of cubic expansion formulas and quadratic factoring helps ensure smooth progress.

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Key points

Use this to factor the sum of two cubes, $a^3 + b^3$ . The first factor is $(a + b)$ , and the middle term of the second factor has a minus sign ( $-ab$ ).

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Use this to factor the difference of two cubes, $a^3 - b^3$ . The first factor is $(a - b)$ , and the middle term of the second factor has a plus sign ( $+ab$ ).

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Note that the middle term in the second factor is $ab$ , not $2ab$ . Additionally, this quadratic factor cannot be factored further over real numbers.

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Practice 1Coming soon