Difference between revisions of "Common factorizations"

Revision as of 03:14, 6 July 2023

These are common factorizations.

Basic Factorizations

$x^2-y^2=(x+y)(x-y)$

$x^3+y^3=(x+y)(x^2-xy+y^2)$

$x^3-y^3=(x-y)(x^2+xy+y^2)$

$x^{2n+1}+y^{2n+1}=(x+y)(x^{2n}-x^{2n-1}y+x^{2n-2}y^2-\ldots-xy^{2n-1}+y^{2n})$

$x^{n}-y^{n}=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-1}+y^n)$

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, useful factorizations.

$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)$

$(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)$

Esoteric Identities

$a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2$

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$

Other Resources

@@ Line 36: / Line 36: @@
 Isn't AoPS supposed to be beyond formula sheets?
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+*[https://artofproblemsolving.com/wiki/index.php/Sum_and_difference_of_powers Sum_and_difference_of_powers]
 [[Category:Algebra]]

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