Difference between revisions of "Common factorizations"

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These are ''common factorizations'' that are used all the timeThese should be memorized, but one should also know how they are derived.
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These are '''common factorizations'''.
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==Basic Factorizations==
 
==Basic Factorizations==
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== Vieta's/Newton Factorizations ==
 
== 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.
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<!-- What exactly do these relations have to do with Vieta's relations? -->
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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, useful factorizations.
  
 
*<math>(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)</math>
 
*<math>(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)</math>
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*<math>(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)</math>
 
*<math>(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)</math>
  
== Advanced Factorizations ==
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== Esoteric Identities ==
*<math>a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2</math>
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*<math>a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2</math> <!-- This isn't a factorization . . . -->
  
 
*<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^2+b^2+c^2-ab-ac-bc)</math>
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== Other Resources ==
 
== Other Resources ==
  
* [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Common Factorizations].
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* [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Factorizations] <!-- Do we really have to link to something like this?
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Isn't AoPS supposed to be beyond formula sheets?
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[[Category:Elementary algebra]]
 
[[Category:Elementary algebra]]

Revision as of 23:02, 3 May 2009

These are common factorizations.

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, useful factorizations.

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

Esoteric Identities

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

Other Resources