# Difference between revisions of "Factoring"

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===Why Factor=== | ===Why Factor=== | ||

− | Factoring equations is an essential part of problem solving. Applying number theory to products yields many results. | + | Factoring equations is an essential part of problem solving. Applying [[Number Theory | number theory]] to products yields many results. |

There are many ways to factor. | There are many ways to factor. | ||

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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 factorizations that show up everywhere. | 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. | ||

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

− | *<math>\displaystyle (a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)</math> | + | *<math>\displaystyle (a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)</math> |

+ | |||

+ | *<math>\displaystyle (a+b+c)^5=a^5+b^5+c^5+5(a+b)(b+c)(c+a)(a^2+b^2+c^2+ab+bc+ca) </math> | ||

== Another Useful Factorization == | == Another Useful Factorization == | ||

<math>a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)</math> | <math>a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)</math> |

## Revision as of 10:28, 21 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.

## Difference of Squares

## Difference of Cubes

## Sum of Cubes

## Simon's Trick

See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on the USAMO or similar national proof contests)

## Summing Series

Also, it is helpful to know how to sum arithmetic series and geometric series.

## 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.

## Another Useful Factorization