Difference between revisions of "Common factorizations"

Latest revision as of 01:06, 28 April 2024

These are common factorizations.

Basic Factorizations

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

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.

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