# Difference between revisions of "Factoring"

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

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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. | 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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*<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> | *<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> | ||

− | == | + | == Other Useful Factorizations == |

− | <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> |

+ | *See [[Simon's Favorite Factoring Trick]] (This is not a recognized formula, please do not quote it on contests) | ||

+ | *[[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> |

## Revision as of 05:05, 23 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 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.

## 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