Binomial Theorem Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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

Solution

  1. 1
    Step 1: Apply (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 with a=xa = x, b=2b = 2.
  2. 2
    Step 2: x3+3x2(2)+3x(4)+8x^3 + 3x^2(2) + 3x(4) + 8.
  3. 3
    Step 3: Simplify: x3+6x2+12x+8x^3 + 6x^2 + 12x + 8.
  4. 4
    Check: (x+2)3(x+2)^3 at x=1x=1: (3)3=27(3)^3 = 27 and 1+6+12+8=271 + 6 + 12 + 8 = 27 ✓

Answer

x3+6x2+12x+8x^3 + 6x^2 + 12x + 8
The Binomial Theorem expands (a+b)n(a+b)^n using coefficients from Pascal's triangle. For n=3n=3, the coefficients are 1, 3, 3, 1.

About Binomial Theorem

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k. Each coefficient (nk)\binom{n}{k} counts the number of ways to choose kk copies of bb from nn factors.

Learn more about Binomial Theorem →

More Binomial Theorem Examples

Example 2 hard

Find the coefficient of [formula] in the expansion of [formula].

Example 3 easy

Expand [formula] using Pascal's triangle.

Example 4 medium

What is [formula]?

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