8.2 Binomial Theorem

8.2 Binomial Theorem

A binomial has two terms (e.g. $a+b$). For positive integer $n$:

Proved by induction using Pascal's identity $\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}$.

Observations:

1. Number of terms $= n+1$.
2. Exponents of $a$ and $b$ in each term sum to $n$.
3. $\binom{n}{r} = \binom{n}{n-r}$ (symmetric coefficients).
4. General term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.

Example 13

Expand $\bigl(\dfrac{a}{2} - \dfrac{2}{a}\bigr)^6$; general term $T_{r+1} = (-1)^r \binom{6}{r} \bigl(\dfrac{a}{2}\bigr)^{6-2r}$.

Example 14

$(9.9)^5 = (10-0.1)^5 \approx 95099.00499$.

Example 15

In $\bigl(\dfrac{3}{2}x - \dfrac{1}{3x}\bigr)^{11}$:

• Term in $x^5$: $T_4 = -\dfrac{40095}{256}x^5$
• Fifth term: $T_5 = \dfrac{4455}{64}x^3$
• Sixth from end: $T_7 = \dfrac{77}{16x}$