Precalculus / Unit 10: Sequences, Series and Intro to Limits
10.4 The Binomial Theorem
Notes
$\displaystyle (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n - k} b^k$, where $\displaystyle \binom{n}{k} = \frac{n!}{k!(n - k)!}$.
- The coefficients come from Pascal's triangle. Row 4: $1, 4, 6, 4, 1$.
- There are $n + 1$ terms. The term with $b^k$ is $\binom{n}{k} a^{n - k} b^k$.
- Example: $(x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16$.
- Watch the signs when $b$ is negative.
Practice
What is $\binom{6}{2}$?
$\displaystyle \frac{6!}{2!\,4!} = \frac{6 \cdot 5}{2} = 15$.
What is the coefficient of $x^3$ in $(x + 2)^5$?
$\binom{5}{2} x^3 \cdot 2^2 = 10 \cdot 4 = 40$.
What is the coefficient of $x^2$ in the expansion of $(2x - 1)^3$?
$\binom{3}{1}(2x)^2(-1) = 3 \cdot 4x^2 \cdot (-1) = -12x^2$.
Ready for more? Take the Unit 10 test.