Binomial theorem

High School

Definition

Also known as "binomial expansion." It is used to describe or expand the binomial expression raised to a power. For example:\( (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\). Expanding this equation one by one would take very long, so a equation was made (as seen in the picture) to help simplify this process by plugging in numbers.

Worked examples

\((x + 2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3 = x^3 + 6x^2 + 12x + 8\)

The coefficients 1, 3, 3, 1 come from Pascal's triangle or the binomial formula; each term decreases in \(x\) and increases in 2.

\((2a - b)^4 = (2a)^4 + 4(2a)^3(-b) + 6(2a)^2(-b)^2 + 4(2a)(-b)^3 + (-b)^4 = 16a^4 - 32a^3b + 24a^2b^2 - 8ab^3 + b^4\)

Signs alternate when the second term is negative; raise each part to its power before multiplying by the coefficient.

Common mistakes

\((a + b)^3 = a^3 + b^3\) → \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) You must include all the middle terms with their binomial coefficients — only the first and last terms are pure powers.
\((x - y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\) → \((x - y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4\) When the second term is negative, odd-power terms flip sign while even-power terms stay positive.
The coefficient of \(a^2b^2\) in \((a+b)^4\) is 4. → The coefficient is 6. Use Pascal's triangle row 4: (1, 4, 6, 4, 1), or calculate \(\binom{4}{2} = 6\).

Where you'll use it next

You'll apply the binomial theorem in probability (binomial distributions), calculus (Taylor series and approximations), and combinatorics. It also streamlines algebraic manipulation in precalculus and higher math.

Found in 1 StudyPug lesson

The Binomial Theorem

Grade 12 Math

Expand (x+y) to any power using the coefficients from row n of Pascal's triangle.

12th Grade12th