Difference between revisions of "Difference of squares"

Latest revision as of 01:27, 25 January 2023

Recognizing a difference of squares is a commonly used factoring technique in algebra. It refers to the identity $a^2 - b^2 = (a+b)(a-b)$ . Note that this identity depends only on the (right and left) distributive property and the commutative property of multiplication and so holds not only for real or complex numbers but also for polynomials, in arithmetic modulo $m$ for any positive integer $m$ , or more generally in any commutative ring. However, due to matrices not being commutative under multiplication, this identity doesn't hold for matrices.

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-Recognizing a '''difference of squares''' is a commonly used [[factoring]] technique in [[algebra]].  It refers to the identity <math>a^2 - b^2 = (a+b)(a-b)</math>.  Note that this identity depends only on the (right and left) [[distributive property]] and the [[commutative property]] of multiplication and so holds not only for [[real]] or [[complex number]]s but also for [[polynomial]]s, in [[modular arithmetic | arithmetic modulo]] <math>m</math> for any [[positive integer]] <math>m</math>, or more generally in any [[commutative ring]].
+Recognizing a '''difference of squares''' is a commonly used [[factoring]] technique in [[algebra]].  It refers to the identity <math>a^2 - b^2 = (a+b)(a-b)</math>.  Note that this identity depends only on the (right and left) [[distributive property]] and the [[commutative property]] of multiplication and so holds not only for [[real]] or [[complex number]]s but also for [[polynomial]]s, in [[modular arithmetic | arithmetic modulo]] <math>m</math> for any [[positive integer]] <math>m</math>, or more generally in any [[commutative ring]]. However, due to [[matrices]] not being commutative under multiplication, this identity doesn't hold for matrices.
 ==See Also==
 * [[Sum and difference of powers]]
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 [[Category:Algebra]]
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