Binomial Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Binomial Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The binomial theorem unifies counting and algebra—each coefficient counts arrangements.

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

Sense of Study hint: Write out Pascal's triangle to the row you need, then match each coefficient with the corresponding powers of a and b.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Expand (a + b)^4 using Pascal's triangle.

Example 2

medium
What is \binom{6}{2}?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

binomial coefficientexponents