Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Common factorizations"

m
(Typesetting, removing the identity that was not a "factorization", and giving a concrete theme to the third section.)
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
These are '''common factorizations'''.
 
These are '''common factorizations'''.
−
<!--
 
−
I think the formatting looks bad.  Bulleted equations just don't look
 
−
good.  Maybe \begin{align*} . . . \end{align*} would be good.  The equations
 
−
should at least be centered.
 
−
-->
 
  
 
==Basic Factorizations==
 
==Basic Factorizations==
−
*<math>x^2-y^2=(x+y)(x-y)</math>
+
<cmath>
−
 
+
\begin{align*}
−
*<math>x^3+y^3=(x+y)(x^2-xy+y^2)</math>
+
\text{\textbullet}&&x^2-y^2&=(x+y)(x-y)\\
−
 
+
\text{\textbullet}&&x^3+y^3&=(x+y)(x^2-xy+y^2)\\
−
*<math>x^3-y^3=(x-y)(x^2+xy+y^2)</math>
+
\text{\textbullet}&&x^3-y^3&=(x-y)(x^2+xy+y^2)\\
 +
\text{\textbullet}&&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})\\
 +
\text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-1}+y^n)
 +
\end{align*}
 +
</cmath>
  
 
== Vieta's/Newton Factorizations ==
 
== Vieta's/Newton Factorizations ==
  
−
<!-- What exactly do these relations have to do with Vieta's relations? -->
 
 
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.
 
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.
  
−
*<math>(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)</math>
+
<cmath>
 +
\begin{align*}
 +
\text{\textbullet}&&a^2+b^2+c^2+2(ab+bc+ac)&=(a+b+c)^2\\
 +
\text{\textbullet}&&(a+b+c)^3-(a^3+b^3+c^3)&=3(a+b)(b+c)(a+c)
 +
\end{align*}
 +
</cmath>
  
−
*<math>(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(a+c)</math>
+
== Circulant Identities ==
  
−
== Esoteric Identities ==
+
<cmath>
−
*<math>a^2+b^2+c^2-ab-ac-bc=((a-b)^2+(b-c)^2+(c-a)^2)/2</math> <!-- This isn't a factorization . . . -->
+
\begin{align*}
 +
\text{\textbullet}&&a^2-b^2&=\det\begin{bmatrix}a&b\\b&a\end{bmatrix}=(a+b)(a-b)\\
 +
\text{\textbullet}&&a^3+b^3+c^3-3abc&=\det\begin{bmatrix}a&b&c\\c&a&b\\b&c&a\end{bmatrix}\\&&&=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)\\
 +
\text{\textbullet}&&a^4 - 4 a^2 b d - 2 a^2 c^2 + 4 a b^2 c + 4 a c d^2 \\\phantom{\text{\textbullet}}&&- b^4 + 2 b^2 d^2 - 4 b c^2 d + c^4 - d^4&=\det\begin{bmatrix}a&b&c&d\\d&a&b&c\\c&d&a&b\\b&c&d&a\end{bmatrix}\\&&&=(a+b+c+d)(a-b+c-d)((a-c)^2+(b-d)^2)
 +
\end{align*}
 +
</cmath>
  
−
*<math>a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)</math>
+
The matrices above are called [https://en.wikipedia.org/wiki/Circulant_matrix circulant matrices]. In general, the determinant of a circulant matrix will be a multiple of the sum of the entries in any of its rows/columns.
  
 
== Other Resources ==
 
== Other Resources ==
  
−
* [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Factorizations] <!-- Do we really have to link to something like this?
+
* [http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet_Reduced.pdf More Factorizations]  
−
Isn't AoPS supposed to be beyond formula sheets?
+
*[https://artofproblemsolving.com/wiki/index.php/Sum_and_difference_of_powers Sum and difference of powers]
−
-->
 
  
−
[[Category:Elementary algebra]]
+
[[Category:Algebra]]

Revision as of 13:13, 6 July 2023

These are common factorizations.

Basic Factorizations

\begin{align*} \text{\textbullet}&&x^2-y^2&=(x+y)(x-y)\\ \text{\textbullet}&&x^3+y^3&=(x+y)(x^2-xy+y^2)\\ \text{\textbullet}&&x^3-y^3&=(x-y)(x^2+xy+y^2)\\ \text{\textbullet}&&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})\\ \text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-1}+y^n) \end{align*}

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.

\begin{align*} \text{\textbullet}&&a^2+b^2+c^2+2(ab+bc+ac)&=(a+b+c)^2\\ \text{\textbullet}&&(a+b+c)^3-(a^3+b^3+c^3)&=3(a+b)(b+c)(a+c) \end{align*}

Circulant Identities

\begin{align*} \text{\textbullet}&&a^2-b^2&=\det\begin{bmatrix}a&b\\b&a\end{bmatrix}=(a+b)(a-b)\\ \text{\textbullet}&&a^3+b^3+c^3-3abc&=\det\begin{bmatrix}a&b&c\\c&a&b\\b&c&a\end{bmatrix}\\&&&=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)\\ \text{\textbullet}&&a^4 - 4 a^2 b d - 2 a^2 c^2 + 4 a b^2 c + 4 a c d^2 \\\phantom{\text{\textbullet}}&&- b^4 + 2 b^2 d^2 - 4 b c^2 d + c^4 - d^4&=\det\begin{bmatrix}a&b&c&d\\d&a&b&c\\c&d&a&b\\b&c&d&a\end{bmatrix}\\&&&=(a+b+c+d)(a-b+c-d)((a-c)^2+(b-d)^2) \end{align*}

The matrices above are called circulant matrices. In general, the determinant of a circulant matrix will be a multiple of the sum of the entries in any of its rows/columns.

Other Resources

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Common_factorizations&oldid=195318"