Binomial Theorem
Description
The Binomial Theorem provides a quick way to expand expressions of the form $(a+b)^n$, where $n$ is a positive integer. Instead of multiplying $(a+b)$ by itself $n$ times manually, the theorem uses **Binomial Coefficients**, denoted as $\binom{n}{k}$ (read as "n choose k"), to predict the coefficients of each term.
The coefficients directly correspond to the rows of **Pascal's Triangle**. For example, for $n=3$, the coefficients are 1, 3, 3, 1, matching the third row of the triangle.
The theorem works by starting with the highest power of $a$ and zero power of $b$, and then for each subsequent term, decreasing the power of $a$ by 1 and increasing the power of $b$ by 1, until you reach $b^n$.
History & Origins
While named after Newton, the pattern of coefficients was known centuries earlier across the world. Pingala (c. 200 BC): An Indian mathematician who described the "Meru Prastara" (Staircase of Mt. Meru), which is essentially Pascal's Triangle, in the context of analyzing Sanskrit poetry meters. Al-Karaji (c. 1000 AD): A Persian mathematician who fully understood the expansion of $(a+b)^n$ for integer powers and provided proofs. Yang Hui (c. 1200 AD): A Chinese mathematician who presented the triangle, which is still called "Yang Hui's Triangle" in China. Isaac Newton (1665): Newton generalized the theorem to allow for non-integer and negative exponents (infinite series), which was a crucial step in the development of calculus.
Proof by Mathematical Induction
We can prove the theorem is true for all integers $n \geq 1$ using induction.
**Base Case ($n=1$):** $(a+b)^1 = a + b$. The formula gives $\binom{1}{0}a^1b^0 + \binom{1}{1}a^0b^1 = 1a + 1b$. It matches.
**Inductive Step:** Assume the formula is true for some integer $n$. We need to show it works for $n+1$.
Consider $(a+b)^{n+1} = (a+b)(a+b)^n$.
Substitute the formula for $(a+b)^n$: $(a+b) \sum \binom{n}{k} a^{n-k}b^k$.
Distribute both $a$ and $b$ into the sum. You get two sums.
Group like terms (terms with the same powers of $a$ and $b$).
Use Pascal's Identity: $\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$.
The resulting coefficients perfectly match the formula for $n+1$. Thus, it is true for all $n$.
Variables
| Symbol | Meaning |
|---|---|
a, b | Terms in the binomial |
n | Exponent (Positive integer) |
k | Term index (from 0 to n) |
nCr | Binomial Coefficient (n choose k) |
Examples
Basic Calculation
Problem: Expand (x+2)³
Solution:
Expanding with Coefficients
Problem: Expand $(2x - 3)^3$.
Solution: $8x^3 - 36x^2 + 54x - 27$
- Identify $a=2x$, $b=-3$, $n=3$.
- Coefficients for $n=3$ are 1, 3, 3, 1.
- Term 1: $1 \cdot (2x)^3 (-3)^0 = 1 \cdot 8x^3 \cdot 1 = 8x^3$.
- Term 2: $3 \cdot (2x)^2 (-3)^1 = 3 \cdot 4x^2 \cdot (-3) = -36x^2$.
- Term 3: $3 \cdot (2x)^1 (-3)^2 = 3 \cdot 2x \cdot 9 = 54x$.
- Term 4: $1 \cdot (2x)^0 (-3)^3 = 1 \cdot 1 \cdot (-27) = -27$.
- Result: $8x^3 - 36x^2 + 54x - 27$.
Approximation
Problem: Estimate $(1.01)^5$ without a calculator.
Solution: ~1.05
- Rewrite as $(1 + 0.01)^5$.
- Use the first two terms of the expansion: $a^n + n a^{n-1}b$.
- $1^5 + 5(1)^4(0.01)$.
- $1 + 5(0.01) = 1 + 0.05 = 1.05$.
- The exact answer is 1.05101..., so this is a very good estimate.
Common Mistakes
The Freshman's Dream
Thinking that $(a+b)^n = a^n + b^n$. This is completely false (unless $n=1$). You MUST include the middle terms (cross-terms). $(a+b)^2 = a^2 + 2ab + b^2$.
Forgetting Negative Signs
If the expression is $(a - b)^n$, treat it as $(a + (-b))^n$. This means the signs will alternate between positive and negative in the expansion.
Real-World Applications
Probability: Coin Flips
The Binomial Theorem is the foundation of the Binomial Distribution in probability. If you flip a coin $n$ times, the number of ways to get exactly $k$ heads is given by the binomial coefficient $\binom{n}{k}$. The term $\binom{n}{k} p^k (1-p)^{n-k}$ gives the exact probability of $k$ successes.
Genetics
In simple Mendelian genetics, if two hybrid parents (Aa) cross, the offspring traits follow binomial probabilities. For multiple traits or polygenic inheritance, the distribution of phenotypes often approximates the expansion of $(a+b)^n$.
Frequently Asked Questions
What is "n choose k"?
It is the number of ways to choose $k$ items from a set of $n$ distinct items, ignoring order. Formula: $\frac{n!}{k!(n-k)!}$.
Does this work for fractional powers?
Yes, but it produces an infinite series (Newton's Generalized Binomial Theorem) instead of a finite polynomial. This is used in calculus for approximations (Taylor Series).