What can polynomial identities apply to beyond just polynomials?

Question

Jaqueline Byrd · Accepted Answer

The difference of squares identity is a polynomial identity that frequently surfaces in various contexts:
\(\displaystyle{a}^{{2}}-{b}^{{2}}={\left({a}-{b}\right)}{\left({a}+{b}\right)}\)
This occurs when we rationalize the denominators.
Consider this example:
\(\displaystyle\frac{{1}}{{{2}+\sqrt{{{3}}}}}\)
\(\displaystyle=\frac{{{2}-\sqrt{{{3}}}}}{{{\left({2}-\sqrt{{{3}}}\right)}{\left({2}+\sqrt{{{3}}}\right)}}}\)
\(\displaystyle=\frac{{{2}-\sqrt{{{3}}}}}{{{2}^{{2}}+{\color{red}{\cancel{{{\color{black}{{\left({2}\right)}\sqrt{{{3}}}}}}}}}-{\color{red}{\cancel{{{\color{black}{\sqrt{{{3}}}{\left({2}\right)}}}}}}}-{\left(\sqrt{{{3}}}\right)}^{{2}}}}\)
\(\displaystyle=\frac{{{2}-\sqrt{{{3}}}}}{{{2}^{{2}}-{\left(\sqrt{{{3}}}\right)}^{{2}}}}\)
\(\displaystyle=\frac{{{2}-\sqrt{{{3}}}}}{{{4}-{3}}}\)
\(\displaystyle={2}-\sqrt{{{3}}}\)
By noticing the differences in the squares pattern, we risk skipping the following step:
\(\displaystyle=\frac{{{2}-\sqrt{{{3}}}}}{{{2}^{{2}}+{\color{red}{\cancel{{{\color{black}{{\left({2}\right)}\sqrt{{{3}}}}}}}}}-{\color{red}{\cancel{{{\color{black}{\sqrt{{{3}}}{\left({2}\right)}}}}}}}-{\left(\sqrt{{{3}}}\right)}^{{2}}}}\)
Or consider this example with a little Complex arithmetic and trigonometric functions:
\(\displaystyle\frac{{1}}{{{\cos{\theta}}+{i}{\sin{\theta}}}}\)
\(\displaystyle=\frac{{{\cos{\theta}}-{i}{\sin{\theta}}}}{{{\left({\cos{\theta}}-{i}{\sin{\theta}}\right)}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}}}\)
\(\displaystyle=\frac{{{\cos{\theta}}-{i}{\sin{\theta}}}}{{{{\cos}^{{2}}\theta}-{i}^{{2}}{{\sin}^{{2}}\theta}}}\)
\(\displaystyle=\frac{{{\cos{\theta}}-{i}{\sin{\theta}}}}{{{{\cos}^{{2}}\theta}+{{\sin}^{{2}}\theta}}}\)
\(\displaystyle={\cos{\theta}}-{i}{\sin{\theta}}\)
At the other end of the scale, this polynomial identity is sometimes useful for mental arithmetic. For example:
\(\displaystyle{97}\cdot{103}={\left({100}-{3}\right)}{\left({100}+{3}\right)}={100}^{{2}}-{3}^{{2}}={10000}-{9}={9991}\)

What can polynomial identities apply to beyond just polynomials?

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