# Difference between revisions of "Factoring"

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* 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>. | ||

− | * Factor <math>x^4 + 1</math> into two polynomials with real coefficients | + | * Factor <math>x^4 + 1</math> into two polynomials with real coefficients. |

+ | * Given that <math>a+b+c=0</math>, prove that <math>abc=\dfrac{a^3+b^3+c^3}{3}</math>. | ||

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

## Revision as of 13:31, 18 May 2009

**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.
- Given that , prove that .