Notes: `(y - 2)^5` Use the Binomial Theorem to expand and simplify the expression.

Sunday, September 20, 2009

$\displaystyle{{\left({y}-{2}\right)}}^{{5}}$ Use the Binomial Theorem to expand and simplify the expression.

You need to use the binomial formula, such that:

$\displaystyle{{\left({x}+{y}\right)}}^{{n}}={\sum_{{{k}={0}}}^{{n}}}{\left(\matrix{{n}\\{k}}\right)}{{x}}^{{{n}-{k}}}{{y}}^{{k}}$

You need to replace y for x, 2 for y and 5 for n, such that:

$\displaystyle{{\left({y}-{2}\right)}}^{{5}}={5}{C}{0}{{\left({y}\right)}}^{{5}}+{5}{C}{1}{{\left({y}\right)}}^{{4}}\cdot{{\left(-{2}\right)}}^{{1}}+{5}{C}{2}{{\left({y}\right)}}^{{3}}\cdot{{\left(-{2}\right)}}^{{2}}+{5}{C}{3}{{y}}^{{2}}\cdot{{\left(-{2}\right)}}^{{3}}+{5}{C}{4}{y}\cdot{{\left(-{2}\right)}}^{{4}}+{5}{C}{5}{{\left(-{2}\right)}}^{{5}}$

By definition, nC0 = nCn = 1, hence $\displaystyle{5}{C}{0}={5}{C}{5}={1}.$

By definition $\displaystyle{n}{C}{1}={n}{C}{\left({n}-{1}\right)}={n}$ , hence $\displaystyle{5}{C}{1}={5}{C}{4}={5}.$

By definition $\displaystyle{n}{C}{2}=\frac{{{n}{\left({n}-{1}\right)}}}{{2}}$ , hence

$\displaystyle{{\left({y}-{2}\right)}}^{{5}}={{y}}^{{5}}-{10}{{y}}^{{4}}+{40}{{y}}^{{3}}-{80}{{y}}^{{2}}+{80}{y}-{32}$

Hence, expanding the complex number using binomial theorem yields the simplified result $\displaystyle{{\left({y}-{2}\right)}}^{{5}}={{y}}^{{5}}-{10}{{y}}^{{4}}+{40}{{y}}^{{3}}-{80}{{y}}^{{2}}+{80}{y}-{32}$

Notes

Sunday, September 20, 2009

$\displaystyle{{\left({y}-{2}\right)}}^{{5}}$ Use the Binomial Theorem to expand and simplify the expression.

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