Cube of a Binomial
Expansion of a binomial raised to the third power.
Formula
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Example
(x+1)³ = x³+3x²+3x+1
How it works
Cubing a binomial means multiplying it by itself three times, so every term from the first bracket must combine with terms from the other two brackets. After collecting like terms, the coefficients 1, 3, 3, and 1 appear because there are exactly that many ways to form each power of a and b. This is a special case of the binomial theorem and gives a quick way to expand without multiplying out all three brackets by hand.
When to use it
Use this formula when expanding expressions like (m+n)^3 or when simplifying algebraic expressions in equations, factoring, or calculus. It is also useful in geometry and physics problems where a quantity changes by a small amount and must be cubed.
Worked example, step by step
Expand (2y - 3)^3 Let a = 2y and b = -3 Use (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = (2y)^3 + 3(2y)^2(-3) + 3(2y)(-3)^2 + (-3)^3 = 8y^3 - 36y^2 + 54y - 27 Final answer: (2y - 3)^3 = 8y^3 - 36y^2 + 54y - 27
Common mistakes to avoid
- Forgetting the middle terms and writing only a^3 + b^3
- Missing the signs when b is negative
- Incorrectly squaring or cubing coefficients such as treating (2y)^3 as 2y^3 instead of 8y^3
Did You Know?
Expanding (a+b)³ gives coefficients 1, 3, 3, 1 — the fourth row of Pascal's triangle.
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