Binomial Theorem

Algebra Progress

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Binomial Theorem

The binomial theorem, and its uses in factoring polynomials.

Author: Jaden Park

Formula

Although taught in school, we'll review it again but in more depth.

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

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

where ${{n} \choose {k}} = \frac{n!}{k!(n - k)!}$ is a binomial coefficient. In other words when $(a + b)^{n}$ is expanded and like terms are collected are the same as the entries in the $n$ th row of Pascal's Triangle.

For example, $(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3},$ with coefficients $1 = {{3} \choose {0}}$ and $3 = {{3} \choose {1}}$

The binomial theorem comes into factoring in certain hyperspecific polynomials.

For example, if a problem has the polynomial

x^{6} + 5x^{5} + 10x^{4} + 10x^{3} + 5x^{2} + x,

you can factor out the $x$ to get

x(x^{5} + 5x^{4} + 10x^{3} + 10x^{2} + 5x + 1) = x(x + 1)^{5}.

It's also pretty useful to know the first few binomial coefficients.

Final Notes and Tips

Not super common like differences and sums of powers, but I would still memorize.

Differences and Sums of Powers

Algebra Progress

Polynomials

Sequence and Series

Miscellaneous

Formula

Final Notes and Tips

Formula

Final Notes and Tips