Expanding with three terms

Japanese school year: Math I

What you learn

Learn how to expand expressions containing three terms, such as the square of a trinomial, by grouping terms effectively. This technique is useful when simplifying complex symmetric algebraic expressions. Ensuring you are already familiar with two-term multiplication formulas will make this topic much easier to understand.

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Key points

Use this when squaring the sum of three terms: $a$ , $b$ , and $c$ . Add the squares of all three terms plus twice the product of each pair ( $2ab$ , $2bc$ , and $2ca$ ).

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

Use this when multiplying three linear factors with variable $x$ and constants $a$ , $b$ , and $c$ . The coefficients correspond to the sum, the pairwise product sum, and the product $abc$ .

(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc

Use this to expand this special product involving three variables: $a$ , $b$ , and $c$ . The intermediate cross-terms cancel out, leaving the sum of the cubes minus $3abc$ .

(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc

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Practice 1Coming soon