Difference between revisions of "Common factorizations"

Revision as of 09:54, 3 May 2009

These are common factorizations that are used all the time. These should be memorized, but one should also know how they are derived.

Basic Factorizations

$x^2-y^2=(x+y)(x-y)$

$x^3+y^3=(x+y)(x^2-xy+y^2)$

$x^3-y^3=(x-y)(x^2+xy+y^2)$

Vieta's/Newton Factorizations

These factorizations are useful for problem that could otherwise be solved by Newton sums or problems that give a polynomial, and ask a question about the roots. Combined with Vieta's formulas, these are excellent factorizations that show up everywhere.

$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)$

$(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)$

Advanced Factorizations

$a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2$

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$

Other Resources

More Common Factorizations.

@@ Line 17: / Line 17: @@
 == Advanced Factorizations ==
+*<math>a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2</math>
 *<math>a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)</math>
-*<math>a^3+b^3+c^3-3abc=(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2)/2</math>
 == Other Resources ==

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