difference of squares

One of the most known and used formulas (http://planetmath.org/Equation) of mathematics is the one concerning the product of sum and difference:

$(a+b)(a-b)=a^2-b^2$ (1)

This form may be used for multiplying any sum of two numbers (terms) by the difference of the same numbers (terms).

In the form

$a^2-b^2=(a+b)(a-b)$ (2)

the formula is used for factoring binomials which are the difference of two squares.

(1) is sometimes called the conjugate rule, especially in articles written in Sweden (in Swedish: konjugatregel).

(1) is an identic equation for all numbers $a,b$ and, more generally, for arbitrary elements $a,b$ of any commutative ring.  Conversely, it is easy to justify that if (1) is true for all elements $a,b$ of a ring, then the ring is commutative.  By the way, $a+b$ and $a-b$ also commute with each other in a non-commutative ring.

Title	difference of squares
Canonical name	DifferenceOfSquares
Date of creation	2013-03-22 17:45:11
Last modified on	2013-03-22 17:45:11
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Topic
Classification	msc 97D99
Classification	msc 26C99
Classification	msc 13A99
Synonym	conjugate rule
Related topic	ConjugationMnemonic
Related topic	ExampleOnSolvingAFunctionalEquation
Related topic	SquareOfSum
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Related topic	FactoringASumOrDifferenceOfTwoCubes
Related topic	Polynomial
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