# Difference between revisions of "Factoring"

(added binomail theorem link, reorganized stuff to more general "other useful factorizations secton") |
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== Vieta's/Newton Factorizations == | == Vieta's/Newton Factorizations == | ||

− | These factorizations are useful for | + | These factorizations are useful for problems 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. |

*<math>\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)</math> | *<math>\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)</math> | ||

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*[[Binomial theorem]] | *[[Binomial theorem]] | ||

== Practice Problems == | == Practice Problems == | ||

− | * Prove that <math>n^2 + 3n + 5</math> is never divisible by 121 for any positive integer <math>{n}</math> | + | * Prove that <math>n^2 + 3n + 5</math> is never divisible by 121 for any positive integer <math>{n}</math>. |

− | * Prove that <math>2222^{5555} + 5555^{2222}</math> is divisible by 7 - USSR Problem Book | + | * Prove that <math>2222^{5555} + 5555^{2222}</math> is divisible by 7. - USSR Problem Book |

− | * Factor <math>(x-y)^3 + (y-z)^3 + (z-x)^3</math> | + | * Factor <math>(x-y)^3 + (y-z)^3 + (z-x)^3</math>. |

== Other Resources == | == Other Resources == | ||

* [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Common Factorizations]. | * [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Common Factorizations]. |

## Revision as of 14:14, 26 June 2006

Note to readers and editers: Please fix up this page by adding in material from Joe's awesome factoring page.

## Contents

### Why Factor

Factoring equations is an essential part of problem solving. Applying number theory to products yields many results.

There are many ways to factor.

## Differences and Sums of Powers

Using the formula for the sum of a geometric sequence, it's easy to derive the more general formula:

Take note of the specific case where **n is odd:**

This also leads to the formula for the sum of cubes,

## Vieta's/Newton Factorizations

These factorizations are useful for problems 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.

## Other Useful Factorizations

- See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)
- Binomial theorem

## Practice Problems

- Prove that is never divisible by 121 for any positive integer .
- Prove that is divisible by 7. - USSR Problem Book
- Factor .