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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a
polynomial expression In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally ...
can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a sum of (repeated) products. During the expansion, simplifications such as grouping of like terms or cancellations of terms may also be applied. Instead of multiplications, the expansion steps could also involve replacing powers of a sum of terms by the equivalent expression obtained from the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
; this is a shortened form of what would happen if the power were treated as a repeated multiplication, and expanded repeatedly. It is customary to reintroduce powers in the final result when terms involve products of identical symbols. Simple examples of polynomial expansions are the well known rules :(x+y)^2=x^2+2xy+y^2 :(x+y)(x-y)=x^2-y^2 when used from left to right. A more general single-step expansion will introduce all products of a term of one of the sums being multiplied with a term of the other: :(a+b+c+d)(x+y+z)=ax+ay+az+bx+by+bz+cx+cy+cz+dx+dy+dz An expansion which involves multiple nested rewrite steps is that of working out a Horner scheme to the (expanded) polynomial it defines, for instance :1+x(-3+x(4+x(0+x(-12+x\cdot 2))))=1-3x+4x^2-12x^4+2x^5. The opposite process of trying to write an expanded polynomial as a product is called polynomial factorization.


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


Expansion of (x+y)n

: 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 In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
(since Pascal's triangle starts with row and column number of 0). For example, when expanding (x+y)^6, the following is obtained: :x^6+x^5y+x^4y^2+x^3y^3+x^2y^4+xy^5+y^6 \,


See also

* Polynomial factorization * Factorization *
Multinomial theorem In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer ...


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Discussion
Review of Algebra: Expansion
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University of Akron The University of Akron is a public university, public research university in Akron, Ohio, United States. It is part of the University System of Ohio. As a STEM fields, STEM-focused institution, it focuses on industries such as polymers, advance ...
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