Difference between revisions of "Common factorizations"
Franklin.vp (talk | contribs) (Typesetting, removing the identity that was not a "factorization", and giving a concrete theme to the third section.) |
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\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^{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- | + | \text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-2}+y^{n-1}) |
\end{align*} | \end{align*} | ||
</cmath> | </cmath> |
Latest revision as of 01:06, 28 April 2024
These are common factorizations.
Contents
[hide]Basic Factorizations
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.
Circulant Identities
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.