Binomial Theorem/Integral Index


Theorem

Let $X$ be one of the standard number systems $\N$, $\Z$, $\Q$, $\R$ or $\C$.

Let $x, y \in X$.

Then:

\(\ds \forall n \in \Z_{\ge 0}: \, \) \(\ds \paren {x + y}^n\) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k\)
\(\ds \) \(=\) \(\ds x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots + \dbinom n n y^n\)

where $\dbinom n k$ is $n$ choose $k$.


Proof

Basis for the Induction

For $n = 0$ we have:

$\ds \paren {x + y}^0 = 1 = \binom 0 0 x^{0 - 0} y^0 = \sum_{k \mathop = 0}^0 \binom 0 k x^{0 - k} y^k$

This is the basis for the induction.


Induction Hypothesis

This is our induction hypothesis:

$\ds \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$


Induction Step

This is our induction step:

\(\ds \paren {x + y}^{n + 1}\) \(=\) \(\ds \paren {x + y} \paren {x + y}^n\)
\(\ds \) \(=\) \(\ds x \sum_{k \mathop = 0}^n \binom n k x^{n - k}y^k + y \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k\) Inductive Hypothesis
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k x^{n + 1 - k} y^k + \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^{k + 1}\)
\(\ds \) \(=\) \(\ds \binom n 0 x^{n + 1} + \sum_{k \mathop = 1}^n \binom n k x^{n + 1 - k} y^k + \binom n n y^{n + 1} + \sum_{k \mathop = 0}^{n - 1} \binom n k x^{n - k} y^{k + 1}\)
\(\ds \) \(=\) \(\ds x^{n + 1} + y^{n + 1} + \sum_{k \mathop = 1}^n \binom n k x^{n + 1 - k} y^k + \sum_{k \mathop = 0}^{n - 1} \binom n k x^{n - k} y^{k + 1}\)
\(\ds \) \(=\) \(\ds \binom {n + 1} 0 x^{n + 1} + \binom {n + 1} {n + 1} y^{n + 1} + \sum_{k \mathop = 1}^n \binom n k x^{n + 1 - k} y^k + \sum_{k \mathop = 1}^n \binom n {k - 1} x^{n + 1 - k} y^k\)
\(\ds \) \(=\) \(\ds \binom {n + 1} 0 x^{n + 1} + \binom {n + 1} {n + 1} y^{n + 1} + \sum_{k \mathop = 1}^n \paren {\binom n k + \binom n {k - 1} } x^{n + 1 - k} y^k\)
\(\ds \) \(=\) \(\ds \binom {n + 1} 0 x^{n + 1} + \binom {n + 1} {n + 1} y^{n + 1} + \sum_{k \mathop = 1}^n \binom {n + 1} k x^{n + 1 - k} y^k\) Pascal's Rule
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^{n + 1} \binom {n + 1} k x^{n + 1 - k} y^k\)

The result follows by the Principle of Mathematical Induction.

$\blacksquare$


Also presented as

The binomial theorem (for integral index) can also be presented in the form:

\(\ds \forall n \in \Z_{\ge 0}: \, \) \(\ds \paren {x + y}^n\) \(=\) \(\ds x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 2} } {3!} x^{n - 3} y^3 + \cdots + y^n\)

which is seen to be the same as the given form by definition of binomial coefficient.


Examples

Cube of Sum

$\paren {x + y}^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3$


Cube of Difference

$\paren {x - y}^3 = x^3 - 3 x^2 y + 3 x y^2 - y^3$


Fourth Power of Sum

$\paren {x + y}^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4$


Fourth Power of Difference

$\paren {x - y}^4 = x^4 - 4 x^3 y + 6 x^2 y^2 - 4 x y^3 + y^4$


Fifth Power of Sum

$\paren {x + y}^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5$


Fifth Power of Difference

$\paren {x - y}^5 = x^5 - 5 x^4 y + 10 x^3 y^2 - 10 x^2 y^3 + 5 x y^4 - y^5$


Sixth Power of Sum

$\paren {x + y}^6 = x^6 + 6 x^5 y + 15 x^4 y^2 + 20 x^3 y^3 + 15 x^2 y^4 + 6 x y^5 + y^6$


Sixth Power of Difference

$\paren {x - y}^6 = x^6 - 6 x^5 y + 15 x^4 y^2 - 20 x^3 y^3 + 15 x^2 y^4 - 6 x y^5 + y^6$


Power of $11$: $11^4$

$11^4 = \left({10 + 1}\right)^4 = 14 \, 641$


Binomial Theorem: $\paren {1 + x}^7$

$\paren {1 + x}^7 = 1 + 7 x + 21 x^2 + 35 x^3 + 35 x^4 + 21 x^5 + 7 x^6 + x^7$


Square Root of 2

$\sqrt 2 = 2 \paren {1 - \dfrac 1 {2^2} - \dfrac 1 {2^5} - \dfrac 1 {2^7} - \dfrac 5 {2^{11} } - \cdots}$


Also known as

The binomial theorem is also known as the binomial formula.


Also see

  • Definition:Binomial Coefficient


Sources

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  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): combination
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  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): binomial series (expansion)
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Binomial Theorem
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): binomial series (expansion)
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Binomial Theorem
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