Binomial Theorem Formula

The Formula

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

When to use: Each term of (a+b)^n picks 'a' or 'b' from each factor. \binom{n}{k} counts how many ways to pick k b's.

Quick Example

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 Coefficients 1,3,3,1 are row 3 of Pascal's triangle.

Notation

\binom{n}{k} = \frac{n!}{k!(n-k)!} is the binomial coefficient ('n choose k'). \sum denotes summation from k = 0 to n.

What This Formula Means

A formula for fully expanding (a + b)^n into a polynomial sum where the coefficients are the binomial coefficients \binom{n}{k}.

Each term of (a+b)^n picks 'a' or 'b' from each factor. \binom{n}{k} counts how many ways to pick k b's.

Formal View

\forall a, b \in \mathbb{R},\; \forall n \in \mathbb{N}: (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k, where \binom{n}{k} = \frac{n!}{k!(n-k)!}.

Worked Examples

Example 1

medium
Expand (x + 2)^3 using the Binomial Theorem.

Solution

  1. 1
    Step 1: Apply (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 with a = x, b = 2.
  2. 2
    Step 2: x^3 + 3x^2(2) + 3x(4) + 8.
  3. 3
    Step 3: Simplify: x^3 + 6x^2 + 12x + 8.
  4. 4
    Check: (x+2)^3 at x=1: (3)^3 = 27 and 1 + 6 + 12 + 8 = 27 ✓

Answer

x^3 + 6x^2 + 12x + 8
The Binomial Theorem expands (a+b)^n using coefficients from Pascal's triangle. For n=3, the coefficients are 1, 3, 3, 1.

Example 2

hard
Find the coefficient of x^3 in the expansion of (2x + 3)^5.

Common Mistakes

  • Writing (a + b)^n = a^n + b^n — this is only true for n = 1; all the middle terms are missing
  • Miscalculating binomial coefficients — \binom{5}{2} = 10, not \frac{5}{2}
  • Forgetting to apply the exponent to BOTH parts of a term — in (2x + 3)^4, the 2 must be raised to the appropriate power along with x

Why This Formula Matters

Enables quick expansion of powers, approximation techniques, and combinatorial identities.

Frequently Asked Questions

What is the Binomial Theorem formula?

A formula for fully expanding (a + b)^n into a polynomial sum where the coefficients are the binomial coefficients \binom{n}{k}.

How do you use the Binomial Theorem formula?

Each term of (a+b)^n picks 'a' or 'b' from each factor. \binom{n}{k} counts how many ways to pick k b's.

What do the symbols mean in the Binomial Theorem formula?

\binom{n}{k} = \frac{n!}{k!(n-k)!} is the binomial coefficient ('n choose k'). \sum denotes summation from k = 0 to n.

Why is the Binomial Theorem formula important in Math?

Enables quick expansion of powers, approximation techniques, and combinatorial identities.

What do students get wrong about Binomial Theorem?

The exponents always sum to n: in \binom{n}{k} a^{n-k} b^k, the powers (n-k)+k = n.

What should I learn before the Binomial Theorem formula?

Before studying the Binomial Theorem formula, you should understand: binomial coefficient, exponents.

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