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Cubic expansions

Expanding expressions

Japanese school year: Math II

What you learn

Learn how to expand cubic expressions efficiently using algebraic formulas such as the cube of a binomial. This calculation is essential for handling higher-degree polynomials and equations. Having a solid understanding of quadratic expansion formulas and combining like terms will help you master this topic smoothly.

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Key points

Use this when expanding the cube of a sum, (a+b)3(a + b)^3. The outer terms are the cubes a3a^3 and b3b^3, while both middle terms have a coefficient of 3.

(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Use this when expanding the cube of a difference, (a−b)3(a - b)^3. It has the same terms as the sum formula, but the signs alternate between positive and negative.

(a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Use this to expand into the sum of cubes, a3+b3a^3 + b^3. Notice that the middle term in the second factor has a minus sign (−ab-ab), allowing intermediate terms to cancel.

(a+b)(a2−ab+b2)=a3+b3(a + b)(a^2 - ab + b^2) = a^3 + b^3

Use this to expand into the difference of cubes, a3−b3a^3 - b^3. Notice that the middle term in the second factor has a plus sign (+ab+ab), allowing intermediate terms to cancel.

(a−b)(a2+ab+b2)=a3−b3(a - b)(a^2 + ab + b^2) = a^3 - b^3

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