Polynomial expansion: Information from Answers.com

For locally approximating a function with a polynomial, see Taylor series.

In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansions of polynomials are obtained by multiplying together their factors, which results in a sum of terms with variables raised to different degrees.

Expansion of a polynomial written in factored form

To multiply two factors, each term of the first factor must be multiplied by each term of the other factor. If both factors are binomials, the FOIL rule can be used, which stands for "First Outer Inner Last," referring to the terms that are multiplied together. For example, expanding

$(x+2)(2x-5)\,$

yields

2 x 2 - 5 x + 4 x - 10 = 2 x 2 - x - 10.

Expansion of (x+y)ⁿ

Main article: Binomial theorem

When expanding $(x + y) n$ , a special relationship exists between the coefficients of the terms when written in order of descending powers of x and ascending powers of y. The coefficients will be the numbers in the (n + 1)th row of Pascal's triangle.

For example, when expanding $(x + y) 6$ , the following is obtained:

${\color{red}1}x^6+{\color{red}6}x^5y+{\color{red}15}x^4y^2+{\color{red}20}x^3y^3+{\color{red}15}x^2y^4+{\color{red}6}xy^5+{\color{red}1}y^6 \,$

How to solve the problem: turn production-sum into sum-production $\prod_{n}\left( \sum_{m}A_{mn}\right) =\sum_{m}\left(\prod_{n}B_{mn}\right)$ , What is $B m n$ in terms of $A m n$ ?

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Polynomial expansion

Expansion of a polynomial written in factored form

Expansion of (x+y)n

See also

Expansion of (x+y)ⁿ