Section 8.4: Binomial Theorem
This section introduces the Binomial Theorem and how to expand expressions of the form \( (a+b)^n \) using combinatorial coefficients.
Example 1: Expanding a Binomial
Expand \( (x + 2)^3 \) using the Binomial Theorem.
Step 1: Identify coefficients using \( \binom{n}{k} \): n=3
Step 2: Expand: \( \binom{3}{0}x^3 2^0 + \binom{3}{1}x^2 2^1 + \binom{3}{2}x^1 2^2 + \binom{3}{3}x^0 2^3 \)
Step 3: Simplify: \( x^3 + 6x^2 + 12x + 8 \)
Example 2: Coefficient of a Term
Find the coefficient of \( x^2 \) in \( (2x + 3)^4 \).
Step 1: Use formula: \( \binom{4}{2} (2x)^2 (3)^{4-2} \)
Step 2: Compute: \( \binom{4}{2} \cdot 4x^2 \cdot 9 = 6 \cdot 4 \cdot 9 = 216 \)
Step 3: Coefficient: 216
Practice Problems
- Expand \( (x + 1)^4 \) using the Binomial Theorem
- Find the coefficient of \( x^3 \) in \( (x + 2)^5 \)
- Expand \( (2a + 3b)^3 \)
- Find the coefficient of \( a^2b^2 \) in \( (a+b)^4 \)
- Expand \( (x - 1)^5 \)