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Question Number 30600    Answers: 0   Comments: 0

let w_k =e^(i((2kπ)/n)) find A= Π_(k=0) ^(n−1) (a +bw_k ).

$${let}\:{w}_{{k}} ={e}^{{i}\frac{\mathrm{2}{k}\pi}{{n}}} \:\:\:\:{find}\:{A}=\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({a}\:+{bw}_{{k}} \:\right). \\ $$

Question Number 30599    Answers: 0   Comments: 1

decompose inside C(x) F= (1/((x−1)(x^n −1))) .

$${decompose}\:{inside}\:{C}\left({x}\right)\:\:{F}=\:\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)\left({x}^{{n}} \:−\mathrm{1}\right)}\:. \\ $$

Question Number 30598    Answers: 0   Comments: 1

prove that it exist one polynomial p/ p(cosx)=cos(nx) find the roots of p(x) .

$${prove}\:{that}\:{it}\:{exist}\:{one}\:{polynomial}\:{p}/ \\ $$$${p}\left({cosx}\right)={cos}\left({nx}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:. \\ $$

Question Number 30597    Answers: 0   Comments: 0

let p(x)=(1+x)^m −e^(2imx) (1−x)^m factorize p(x) inside C[x].

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{x}\right)^{{m}} \:−{e}^{\mathrm{2}{imx}} \left(\mathrm{1}−{x}\right)^{{m}} \:{factorize}\:{p}\left({x}\right) \\ $$$${inside}\:{C}\left[{x}\right]. \\ $$

Question Number 30596    Answers: 0   Comments: 0

find all polynomial wich verify p(x^2 ) +p(x)p(x+1)=0.

$${find}\:{all}\:{polynomial}\:{wich}\:{verify}\: \\ $$$${p}\left({x}^{\mathrm{2}} \right)\:+{p}\left({x}\right){p}\left({x}+\mathrm{1}\right)=\mathrm{0}. \\ $$

Question Number 30595    Answers: 0   Comments: 1

let f(x)= (1/(x^2 −2cosαx+1)) find f^((n)) .

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} \:−\mathrm{2}{cos}\alpha{x}+\mathrm{1}}\:\:{find}\:{f}^{\left({n}\right)} . \\ $$

Question Number 30594    Answers: 0   Comments: 0

let p(x)=x^3 +1 and q(x)=x^4 +1 prove that D(p,q)=1.

$${let}\:{p}\left({x}\right)={x}^{\mathrm{3}} \:+\mathrm{1}\:{and}\:{q}\left({x}\right)={x}^{\mathrm{4}} \:+\mathrm{1}\:{prove}\:{that} \\ $$$${D}\left({p},{q}\right)=\mathrm{1}. \\ $$

Question Number 30593    Answers: 1   Comments: 0

factorize inside C[x] p(x)=(1+i(x/n))^n −(1−i(x/n))^n .

$${factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right)=\left(\mathrm{1}+{i}\frac{{x}}{{n}}\right)^{{n}} \:−\left(\mathrm{1}−{i}\frac{{x}}{{n}}\right)^{{n}} . \\ $$

Question Number 30592    Answers: 1   Comments: 0

let p(x)=x^(2n) −2cosα x^n +1 1) find roots lf p(x) 2)factorize p(x) inside C[x] 3)factorize p(x) inside R[x].

$${let}\:{p}\left({x}\right)={x}^{\mathrm{2}{n}} \:−\mathrm{2}{cos}\alpha\:{x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{lf}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{R}\left[{x}\right]. \\ $$

Question Number 30590    Answers: 1   Comments: 0

decompose sur R[x] x^(2n+1) −1.

$${decompose}\:{sur}\:{R}\left[{x}\right]\:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:−\mathrm{1}. \\ $$

Question Number 30589    Answers: 0   Comments: 0

let U_n = {z∈C / z^n =1} find S= Σ_(z∈U_n ) (z/((x−z)^2 )) .

$${let}\:{U}_{{n}} =\:\left\{{z}\in{C}\:/\:{z}^{{n}} =\mathrm{1}\right\}\:\:{find} \\ $$$${S}=\:\sum_{{z}\in{U}_{{n}} } \:\:\frac{{z}}{\left({x}−{z}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 30588    Answers: 0   Comments: 0

(n_k )_(1≤k≤n) is a family of integrs numbers let put p(x)=Σ_(k=1) ^n x^n_k and q(x)= Σ_(j=0) ^(n−1) x^j if n_k ≡k−1[n] prove that q divide p.

$$\left({n}_{{k}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} \:{is}\:{a}\:{family}\:{of}\:{integrs}\:{numbers}\:{let}\:{put} \\ $$$${p}\left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:{x}^{{n}_{{k}} } \:\:\:{and}\:{q}\left({x}\right)=\:\sum_{{j}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{x}^{{j}} \: \\ $$$${if}\:{n}_{{k}} \equiv{k}−\mathrm{1}\left[{n}\right]\:{prove}\:{that}\:{q}\:{divide}\:{p}. \\ $$

Question Number 30587    Answers: 0   Comments: 0

find Σ_(k=0) ^(n−1) (−1)^k cos^n (((kπ)/n)).

$${find}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\left(−\mathrm{1}\right)^{{k}} \:{cos}^{{n}} \left(\frac{{k}\pi}{{n}}\right). \\ $$

Question Number 30586    Answers: 0   Comments: 0

let p=1+x+x^2 +....+x^(2^(n+1) −1) and q= 1+x^2^n find α= (p/q) .

$${let}\:{p}=\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+....+{x}^{\mathrm{2}^{{n}+\mathrm{1}} −\mathrm{1}} \:{and}\:\:{q}=\:\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \\ $$$${find}\:\alpha=\:\frac{{p}}{{q}}\:. \\ $$

Question Number 30585    Answers: 0   Comments: 0

find F_n (x)= ∫_0 ^∞ (x^n /(e^(x+n) +1))dx .

$${find}\:\:{F}_{{n}} \left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{{n}} }{{e}^{{x}+{n}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 30584    Answers: 0   Comments: 0

find I= ∫_(−∞) ^(+∞) (e^(−x^2 ) /(a^2 +(v−x)^2 ))dx.

$${find}\:\:{I}=\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{e}^{−{x}^{\mathrm{2}} } }{{a}^{\mathrm{2}} \:+\left({v}−{x}\right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30583    Answers: 0   Comments: 0

decompose F =(1/((x^2 −1)^n )) inside C[x].n from N.

$${decompose}\:{F}\:=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{{n}} }\:{inside}\:{C}\left[{x}\right].{n}\:{from}\:{N}. \\ $$

Question Number 30582    Answers: 0   Comments: 0

x_1 , x_2 , x_(3 ) are roots of the polynomial x^3 −x+1 find the polynomial wich have for roots x_1 ^3 ,x_2 ^3 and x_3 ^3 .

$${x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}\:} \:{are}\:{roots}\:{of}\:{the}\:{polynomial}\:{x}^{\mathrm{3}} \:−{x}+\mathrm{1}\:{find} \\ $$$${the}\:{polynomial}\:{wich}\:{have}\:{for}\:{roots}\:{x}_{\mathrm{1}} ^{\mathrm{3}} \:,{x}_{\mathrm{2}} ^{\mathrm{3}} \:{and}\:{x}_{\mathrm{3}} ^{\mathrm{3}} \:\:. \\ $$

Question Number 30581    Answers: 0   Comments: 0

decompose inside C[x] F= (1/((x+iy)^n )) .

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\:\frac{\mathrm{1}}{\left({x}+{iy}\right)^{{n}} }\:. \\ $$

Question Number 30580    Answers: 0   Comments: 1

decompose inside C[x] F= (x^n /(x^m +1)) with m≥n+2 then find ∫_0 ^∞ (x^n /(x^m +1))dx.

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}\:{with}\:{m}\geqslant{n}+\mathrm{2} \\ $$$${then}\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}{dx}. \\ $$

Question Number 30579    Answers: 0   Comments: 0

decompose inside C[x] F= ((x^n −1)/(x^(2n) −1)) .

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\frac{{x}^{{n}} −\mathrm{1}}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:. \\ $$

Question Number 30578    Answers: 0   Comments: 0

decompose F(x)= (1/((x^2 +1)^n )) on C[x].with n fromN.

$${decompose}\:{F}\left({x}\right)=\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{n}} }\:{on}\:{C}\left[{x}\right].{with}\:{n}\:{fromN}. \\ $$

Question Number 30576    Answers: 0   Comments: 1

let consider the equation x^3 +px +q find S= Σ_(i≠j) (x_i /x_j ) .

$${let}\:{consider}\:{the}\:{equation}\:{x}^{\mathrm{3}} \:+{px}\:+{q}\: \\ $$$${find}\:{S}=\:\sum_{{i}\neq{j}} \:\frac{{x}_{{i}} }{{x}_{{j}} }\:. \\ $$

Question Number 30575    Answers: 0   Comments: 0

find ∫∫_D (x^2 +y^2 )dxdy with D={(x,y)/ x≤1 and x^2 ≤y≤2 }.

$${find}\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}=\left\{\left({x},{y}\right)/\:{x}\leqslant\mathrm{1}\:{and}\:{x}^{\mathrm{2}} \leqslant{y}\leqslant\mathrm{2}\:\right\}. \\ $$

Question Number 30574    Answers: 0   Comments: 0

find ∫∫_([1,e]^2 ) ln(xy)dxdy.

$${find}\:\int\int_{\left[\mathrm{1},{e}\right]^{\mathrm{2}} } \:\:\:{ln}\left({xy}\right){dxdy}. \\ $$

Question Number 30573    Answers: 0   Comments: 0

find ∫∫_([0,1]×[0,1]) (x^2 /(1+y^2 ))dxdy.

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right]} \:\:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{y}^{\mathrm{2}} }{dxdy}. \\ $$

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