# Difference between revisions of "Factoring"

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− | Factoring is an essential part of | + | '''Factoring''' is an essential part of any mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious. |

==Differences and Sums of Powers== | ==Differences and Sums of Powers== | ||

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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> | ||

− | Using the formula for the sum of a [[ | + | Using the formula for the sum of a [[geometric sequence]], it's easy to derive the more general formula: |

<math>a^n-b^n=(a-b)(a^{n-1}+ba^{n-2} + \cdots + b^{n-2}a + b^{n-1})</math> | <math>a^n-b^n=(a-b)(a^{n-1}+ba^{n-2} + \cdots + b^{n-2}a + b^{n-1})</math> | ||

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== Other Useful Factorizations == | == 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> | ||

− | * | + | * [[Simon's Favorite Factoring Trick]] |

− | *[[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>. |

## Revision as of 12:37, 20 July 2006

**Factoring** is an essential part of any mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an integer or polynomial) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious.

## Contents

## 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

## Practice Problems

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