# Difference between revisions of "Factoring"

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

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

− | *<math> | + | *<math>(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 == | == 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> | ||

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== 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]. | ||

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+ | [[Category:Definition]] | ||

+ | [[Category:Elementary algebra]] |

## Revision as of 17:04, 25 December 2007

**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:

In addition, if is odd:

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

Another way to discover these factorizations is the following: the expression is equal to zero if . If one factorizes a product which is equal to zero, one of the factors must be equal to zero, so must have a factor of . Similarly, we note that the expression when is odd is equal to zero if , so it must have a factor of . Note that when is even, , rather than 0, so this gives us no useful information.

## 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 .
- Factor into two polynomials with real coefficients