Difference between revisions of "Binomial Theorem"

Revision as of 18:21, 22 April 2008

The Binomial Theorem states that for real or complex $a$ , $b$ , and non-negative integer $n$ ,

$(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$

This may be easily shown for the integers: $(a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}$ . Repeatedly using the distributive property, we see that for a term $a^m b^{n-m}$ , we must choose $m$ of the $n$ terms to contribute an $a$ to the term, and then each of the other $n-m$ terms of the product must contribute a $b$ . Thus, the coefficient of $a^m b^{n-m}$ is $\binom{m}{n}$ . Extending this to all possible values of $m$ from $0$ to $n$ , we see that $(a+b)^n = \sum_{k=0}^{n}{\binom{n}{k}}\cdot a^k\cdot b^{n-k}$ .

Generalization

The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents. For any real or complex $a$ , $b$ , and $r$ ,

$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$

Usage

Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial $x^5+4x^4+6x^3+4x^2+x$ , one could factor it as such: $x(x^4+4x^3+6x^2+4x+1)=x(x+1)^{4}$ . It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.

@@ Line 1: / Line 1: @@
-==The Theorem==
+The '''Binomial Theorem''' states that for [[real]] or [[complex]] <math>a</math>, <math>b</math>, and [[non-negative]] [[integer]] <math>n</math>,
-First discovered by [[Isaac Newton]], the '''Binomial Theorem''' states that for [[real number | real]] or [[complex number |complex]] ''a'',''b'',<br><math>(a+b)^n = \sum_{k=0}^{n}{\binom{n}{k}}\cdot a^k\cdot b^{n-k}</math>.
+<center><math>(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k</math></center>
-This may be easily shown for the [[integer]]s:<br>
+This may be easily shown for the [[integer]]s: <math>(a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math>. Repeatedly using the [[distributive property]], we see that for a term <math>a^m b^{n-m}</math>, we must choose <math>m</math> of the <math>n</math> terms to contribute an <math>a</math> to the term, and then each of the other <math>n-m</math> terms of the product must contribute a <math>b</math>. Thus, the coefficient of <math>a^m b^{n-m}</math> is <math>\binom{m}{n}</math>. Extending this to all possible values of <math>m</math> from <math>0</math> to <math>n</math>, we see that <math>(a+b)^n = \sum_{k=0}^{n}{\binom{n}{k}}\cdot a^k\cdot b^{n-k}</math>.
-<math>(a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math>.
-<br>Repeatedly using the [[distributive property]], we see that for a term <math>a^m b^{n-m}</math>, we must choose <math>m</math> of the <math>n</math> terms to contribute an <math>a</math> to the term, and then each of the other <math>n-m</math> terms of the product must contribute a <math>b</math>. Thus, the coefficient of <math>a^m b^{n-m}</math> is <math>\binom{m}{n}</math>. Extending this to all possible values of <math>m</math> from <math>0</math> to <math>n</math>, we see that <math>(a+b)^n = \sum_{k=0}^{n}{\binom{n}{k}}\cdot a^k\cdot b^{n-k}</math>.
+==Generalization==
+The Binomial Theorem was generalized by [[Isaac Newton]], who used an [[infinite]] [[series]] to allow for complex [[exponent]]s. For any [[real]] or [[complex]] <math>a</math>, <math>b</math>, and <math>r</math>,
+<center><math>(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k</math></center>
 ==Usage==
@@ Line 14: / Line 16: @@
 [[Category:Theorems]]
+[[Category:Number Theory]]

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Difference between revisions of "Binomial Theorem"

Revision as of 18:21, 22 April 2008

Generalization

Usage

See also