Difference between revisions of "Factoring"

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.

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

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)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 $n^2 + 3n + 5$ 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

More Common Factorizations.

@@ Line 24: / Line 24: @@
 <math>a^3+b^3=(a+b)(a^2-ab+b^2)</math>
-==Simon's Trick==
-See [[Simon's Favorite Factoring Trick]]
-(This is not a recognized formula, please do not quote it on the USAMO or similar national proof contests)
-==Summing Sequences==
-Also, it is helpful to know how to sum [[arithmetic sequence]] and [[geometric sequence]].
 == 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.
@@ Line 37: / Line 33: @@
 *<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>
-== Another Useful Factorization ==
+== 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 ==
 * Prove that <math>n^2 + 3n + 5</math> is never divisible by 121 for any positive integer <math>{n}</math>

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