Common factorizations

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These are common factorizations.

Basic Factorizations

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

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

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