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Category: Algebra

f-x-16-x-4-16-x-faind-volue-of-f-1-20-f-2-20-f-19-20-

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Question Number 171189 by mathlove last updated on 09/Jun/22 $${f}\left({x}\right)=\frac{\mathrm{16}^{{x}} }{\mathrm{4}+\mathrm{16}^{{x}} }\:\:\:\:\:{faind}\:{volue}\:{of} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{20}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{20}}\right)+……..+{f}\left(\frac{\mathrm{19}}{\mathrm{20}}\right) \\ $$ Answered by cortano1 last updated on 09/Jun/22 $$\:{f}\left({x}\right)=\frac{\mathrm{16}^{{x}} +\mathrm{4}−\mathrm{4}}{\mathrm{16}^{{x}}…

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prove-that-m-m-n-n-k-m-m-1-n-n-k-1-m-m-2-n-n-k-2-m-m-k-n-n-m-n-k-

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Question Number 105625 by  M±th+et+s last updated on 30/Jul/20 $${prove}\:{that} \\ $$$$\begin{pmatrix}{{m}}\\{{m}}\end{pmatrix}\begin{pmatrix}{\:\:\:{n}}\\{{n}−{k}}\end{pmatrix}+\begin{pmatrix}{\:\:\:\:{m}}\\{{m}−\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\:\:\:\:\:\:\:{n}}\\{{n}−{k}+\mathrm{1}}\end{pmatrix} \\ $$$$+\begin{pmatrix}{\:\:\:{m}}\\{{m}−\mathrm{2}}\end{pmatrix}\begin{pmatrix}{\:\:\:\:\:\:\:\:{n}}\\{{n}−{k}+\mathrm{2}}\end{pmatrix}+……+\begin{pmatrix}{\:\:\:\:{m}}\\{{m}−{k}}\end{pmatrix}\begin{pmatrix}{{n}}\\{{n}}\end{pmatrix} \\ $$$$=\begin{pmatrix}{{m}+{n}}\\{\:\:\:\:\:{k}}\end{pmatrix} \\ $$ Answered by prakash jain last updated on…

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let-p-be-a-non-constant-polynomial-with-degree-n-such-that-p-z-a-n-z-n-a-0-show-that-if-z-C-with-z-max-1-2-a-n-i-1-n-1-a-i-then-1-2-a-n-z-n-p-z-3-2-a-n-z-n-i

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Question Number 171145 by floor(10²Eta[1]) last updated on 08/Jun/22 $$\mathrm{let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{non}\:\mathrm{constant}\:\mathrm{polynomial}\:\mathrm{with}\: \\ $$$$\mathrm{degree}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}:\:\mathrm{p}\left(\mathrm{z}\right)=\mathrm{a}_{\mathrm{n}} \mathrm{z}^{\mathrm{n}} +…+\mathrm{a}_{\mathrm{0}} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{z}\in\mathbb{C}\:\mathrm{with}\mid\mathrm{z}\mid\geqslant\mathrm{max}\left\{\mathrm{1},\frac{\mathrm{2}}{\mid\mathrm{a}_{\mathrm{n}} \mid}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mid\mathrm{a}_{\mathrm{i}} \mid\right\} \\ $$$$\mathrm{then}\:\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \leqslant\mid\mathrm{p}\left(\mathrm{z}\right)\mid\leqslant\frac{\mathrm{3}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}}…

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1-find-the-roots-of-p-x-1-ix-x-2-n-1-ix-x-2-n-with-n-integr-natural-2-factorize-p-x-inside-C-x-3-give-p-x-at-form-a-p-x-p-

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Question Number 40042 by abdo mathsup 649 cc last updated on 15/Jul/18 $$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:\: \\ $$$${p}\left({x}\right)=\left(\mathrm{1}+{ix}\:+{x}^{\mathrm{2}} \right)^{{n}} −\left(\mathrm{1}−{ix}+{x}^{\mathrm{2}} \right)^{{n}} \:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:\:{C}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{p}\left({x}\right)\:{at}\:{form}\:\:\Sigma\:{a}_{{p}}…

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Question-171109

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Question Number 171109 by akolade last updated on 08/Jun/22 Commented by Rasheed.Sindhi last updated on 09/Jun/22 $$\underset{\left(\mathrm{1}\right)} {\underbrace{{a}+{b}=\mathrm{2}}},\:\underset{\left({ii}\right)} {\underbrace{\mathrm{3}{a}^{\mathrm{2}} +\mathrm{3}{b}^{\mathrm{2}} =\mathrm{4}}},\:{a}^{\mathrm{2019}} +{b}^{\mathrm{2019}} =? \\ $$$$\left({i}\right)\Rightarrow\frac{{a}}{{b}}+\mathrm{1}=\frac{\mathrm{2}}{{b}}\Rightarrow\begin{array}{|c|}{\frac{{a}}{{b}}=\frac{\mathrm{2}−{b}}{{b}}….\left({A}\right)}\\\hline\end{array}…

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