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Difference between revisions of "Factoring"

(→‎Differences and Sums of Powers)
(added binomail theorem link, reorganized stuff to more general "other useful factorizations secton")
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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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==Simon's Trick==
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See [[Simon's Favorite Factoring Trick]]
 
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(This is not a recognized formula, please do not quote it on the USAMO or similar national proof contests)
 
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==Summing Sequences==
 
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Also, it is helpful to know how to sum [[arithmetic sequence]] and [[geometric sequence]].
 
 
== 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>
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== Another Useful Factorization ==
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== Other Useful Factorizations ==
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<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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*<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*See [[Simon's Favorite Factoring Trick]] (This is not a recognized formula, please do not quote it on contests)
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*[[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 04:05, 23 June 2006

Note to readers and editers: Please fix up this page by adding in material from Joe's awesome factoring page.


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

$a^2-b^2=(a+b)(a-b)$

$a^3-b^3=(a-b)(a^2+ab+b^2)$

Using the formula for the sum of a geometric sequence, it's easy to derive the more general formula:

$a^n-b^n=(a-b)(a^{n-1}+ba^{n-2} + \cdots + b^{n-2}a + b^{n-1})$

Take note of the specific case where n is odd:

$a^n+b^n=(a+b)(a^{n-1} - ba^{n-2} + b^2a^{n-3} - b^3a^{n-4} + \cdots + b^{n-1})$

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

$a^3+b^3=(a+b)(a^2-ab+b^2)$


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.

  • $\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$
  • $\displaystyle (a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)$
  • $\displaystyle (a+b+c)^5=a^5+b^5+c^5+5(a+b)(b+c)(c+a)(a^2+b^2+c^2+ab+bc+ca)$

Other Useful Factorizations

Practice Problems

  • Prove that $n^2 + 3n + 5$ is never divisible by 121 for any positive integer ${n}$
  • Prove that $2222^{5555} + 5555^{2222}$ is divisible by 7 - USSR Problem Book
  • Factor $(x-y)^3 + (y-z)^3 + (z-x)^3$

Other Resources

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