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[parent] difference of squares (Topic)

One of the most known and used formulas of mathematics is the one concerning the product of sum and difference:

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

$\displaystyle a^2-b^2 = (a+b)(a-b)$ (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.



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See Also: conjugation (mnemonic), example of solving a functional equation, square of sum, grouping method for factoring polynomials, incircle radius determined by Pythagorean triple, factoring a sum or difference of two cubes, polynomial

Other names:  conjugate rule

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Cross-references: non-commutative, commutative, ring, commutative ring, equation, squares, binomials, difference, sum, product
There are 2 references to this entry.

This is version 7 of difference of squares, born on 2008-01-20, modified 2008-03-27.
Object id is 10204, canonical name is DifferenceOfSquares.
Accessed 581 times total.

Classification:
AMS MSC: 13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous)
 26C99 (Real functions :: Polynomials, rational functions :: Miscellaneous)
 97D99 (Mathematics education :: Education and instruction in mathematics :: Miscellaneous)
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