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1) decompose F(x)=(1/((x−a)^n (x−b)^n )) witha≠b 2) find the values of Σ_(k=0) ^(n−1) (C_(p+k−1) ^k /2^(k+p) ) + Σ_(k=0) ^(p−1) (C_(n+k−1) ^k /2^(k+n) )

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)=\frac{\mathrm{1}}{\left({x}−{a}\right)^{{n}} \left({x}−{b}\right)^{{n}} } \\ $$$${witha}\neq{b} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{{C}_{{p}+{k}−\mathrm{1}} ^{{k}} }{\mathrm{2}^{{k}+{p}} }\: \\ $$$$+\:\sum_{{k}=\mathrm{0}} ^{{p}−\mathrm{1}} \:\:\:\frac{{C}_{{n}+{k}−\mathrm{1}} ^{{k}} }{\mathrm{2}^{{k}+{n}} } \\ $$

Question Number 50379 Answers: 0 Comments: 0

let U_n ={z∈C /z^n =1} find two polynom A(x) and B(x) verify Σ_(w∈U_n ) (w/((x−w)^2 )) =((A(x))/(B(x)))

$${let}\:{U}_{{n}} =\left\{{z}\in{C}\:/{z}^{{n}} =\mathrm{1}\right\}\:\:{find}\:{two}\:{polynom}\:{A}\left({x}\right) \\ $$$${and}\:{B}\left({x}\right)\:{verify}\:\:\sum_{{w}\in{U}_{{n}} } \:\:\:\:\frac{{w}}{\left({x}−{w}\right)^{\mathrm{2}} }\:=\frac{{A}\left({x}\right)}{{B}\left({x}\right)} \\ $$

Question Number 50377 Answers: 0 Comments: 0

let p is a polynome with degp=n≥2 hsving n roots simples prove that Σ_(k=1) ^n (1/(p^, (x_k ))) =0

$${let}\:{p}\:{is}\:{a}\:{polynome}\:{with}\:{degp}={n}\geqslant\mathrm{2}\:{hsving}\:{n}\:{roots} \\ $$$${simples}\:{prove}\:{that}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{p}^{,} \left({x}_{{k}} \right)}\:=\mathrm{0} \\ $$

Question Number 50376 Answers: 0 Comments: 0

let p∈ K[x] prove that p−x divide pop(x)−x

$${let}\:{p}\in\:{K}\left[{x}\right]\:{prove}\:{that}\:{p}−{x}\:{divide}\:{pop}\left({x}\right)−{x} \\ $$

Question Number 50375 Answers: 1 Comments: 0

let f(x)=(1/(cosx)) prove that f^()n)) (x)=((p_n (sinx))/(cos^(n+1) x)) with p_n is apolynom 2) calculate p_1 ,p_2 and p_3 3) detdrmine p_n (1).

$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}}\:{prove}\:{that}\:{f}^{\left.\right)\left.{n}\right)} \left({x}\right)=\frac{{p}_{{n}} \left({sinx}\right)}{{cos}^{{n}+\mathrm{1}} {x}} \\ $$$${with}\:{p}_{{n}} \:{is}\:{apolynom} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} \:{and}\:{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{3}\right)\:{detdrmine}\:{p}_{{n}} \left(\mathrm{1}\right). \\ $$

Question Number 50374 Answers: 0 Comments: 0

determine all polynoms p ∈R[x] wich verify p(x^2 )=p(x)p(x+1)

$${determine}\:{all}\:{polynoms}\:{p}\:\in{R}\left[{x}\right]\:{wich}\:{verify} \\ $$$${p}\left({x}^{\mathrm{2}} \right)={p}\left({x}\right){p}\left({x}+\mathrm{1}\right) \\ $$

Question Number 50373 Answers: 0 Comments: 0

decompose in prime factors the polynom p=x^(2n) −2cosα x^n +1

$${decompose}\:{in}\:{prime}\:{factors}\:{the}\:{polynom} \\ $$$${p}={x}^{\mathrm{2}{n}} −\mathrm{2}{cos}\alpha\:{x}^{{n}} \:+\mathrm{1} \\ $$

Question Number 50372 Answers: 0 Comments: 0

prove that 2^(n+1) divide[(1+(√3))^(2n+1) ] for all n integr natural.

$${prove}\:{that}\:\mathrm{2}^{{n}+\mathrm{1}} \:{divide}\left[\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}+\mathrm{1}} \right]\:{for}\:{all}\:{n} \\ $$$${integr}\:{natural}. \\ $$

Question Number 50371 Answers: 0 Comments: 0

let F_n =2^2^n +1 (fermat numbers) prove that Δ(F_m ,F_n )=1 for m≠n

$${let}\:{F}_{{n}} =\mathrm{2}^{\mathrm{2}^{{n}} } \:+\mathrm{1}\:\:\:\:\left({fermat}\:{numbers}\right) \\ $$$${prove}\:{that}\:\Delta\left({F}_{{m}} ,{F}_{{n}} \right)=\mathrm{1}\:{for}\:{m}\neq{n} \\ $$

Question Number 50370 Answers: 0 Comments: 0

prove that ∀ (x,y)∈Z^2 x^(19) y−xy^(19) is divided by 798.

$${prove}\:{that}\:\forall\:\left({x},{y}\right)\in{Z}^{\mathrm{2}} \\ $$$${x}^{\mathrm{19}} {y}−{xy}^{\mathrm{19}} \:{is}\:{divided}\:{by}\:\mathrm{798}. \\ $$

Question Number 50369 Answers: 1 Comments: 0

find x ,y from Z wich verify y^2 =x(x+1)(x+7)(x+8)

$${find}\:{x}\:,{y}\:{from}\:{Z}\:\:{wich}\:{verify} \\ $$$${y}^{\mathrm{2}} ={x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{7}\right)\left({x}+\mathrm{8}\right) \\ $$

Question Number 50368 Answers: 0 Comments: 1

find all (x,y)∈Q^(+★^2 ) and x^y =y^x and x<y

$${find}\:{all}\:\left({x},{y}\right)\in{Q}^{+\bigstar^{\mathrm{2}} } \:\:{and}\:\:{x}^{{y}} ={y}^{{x}} \:\:{and}\:{x}<{y} \\ $$

Question Number 50367 Answers: 0 Comments: 0

calculate Σ_(k=p) ^(2p) (C_k ^p /2^k )

$${calculate}\:\sum_{{k}={p}} ^{\mathrm{2}{p}} \:\:\frac{{C}_{{k}} ^{{p}} }{\mathrm{2}^{{k}} } \\ $$

Question Number 50366 Answers: 0 Comments: 0

let A = (((0 m m^2 )),(((1/m) 0 m)) ) ((1/m^2 ) (1/m) 0 ) A ∈ M_3 (R) and m not 0 1) find relation betwen I_3 , A and A^2 2) is A inversible .determine A^(−1) in case of exist 3) find the propers values of A.

$${let}\:\:{A}\:=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:{m}\:\:\:\:\:\:{m}^{\mathrm{2}} }\\{\frac{\mathrm{1}}{{m}}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:{m}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{1}}{{m}^{\mathrm{2}} }\:\:\:\:\frac{\mathrm{1}}{{m}}\:\:\:\:\:\:\mathrm{0}\:\:\:\right) \\ $$$${A}\:\in\:{M}_{\mathrm{3}} \left({R}\right)\:\:{and}\:{m}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{relation}\:{betwen}\:{I}_{\mathrm{3}} ,\:{A}\:{and}\:{A}^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{is}\:{A}\:{inversible}\:\:\:.{determine}\:{A}^{−\mathrm{1}} \:{in}\:{case}\:{of}\:{exist} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{propers}\:{values}\:{of}\:{A}. \\ $$

Question Number 50365 Answers: 0 Comments: 0

let J = [((1 1 1)),((1 1 1)) ] [((1 1 1)),() ] element of M_3 (R) find J^n

$${let}\:{J}\:=\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{}\end{bmatrix} \\ $$$${element}\:{of}\:{M}_{\mathrm{3}} \left({R}\right) \\ $$$${find}\:{J}^{{n}} \\ $$$$ \\ $$

Question Number 50364 Answers: 0 Comments: 0

let A = (((2 1)),((1 2)) ) calculate A^n

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix} \\ $$$${calculate}\:{A}^{{n}} \\ $$

Question Number 50363 Answers: 0 Comments: 0

let p ∈K_n [x] snd A and H two rlements of K[x] 1) prove that p(A(x)+H(x))=Σ_(k=0) ^n ((p^((k)) (A(x)))/(k!)).(H(x))^k 2)find the condition that p(A(x)+H(x))is divided by H(x)≠0 3) if p(x)≠c prove that p(p(x))−x is divided by p(x)−x.

$${let}\:{p}\:\in{K}_{{n}} \left[{x}\right]\:{snd}\:{A}\:{and}\:{H}\:{two}\:{rlements}\:{of}\:{K}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{p}\left({A}\left({x}\right)+{H}\left({x}\right)\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \frac{{p}^{\left({k}\right)} \left({A}\left({x}\right)\right)}{{k}!}.\left({H}\left({x}\right)\right)^{{k}} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{condition}\:{that}\:{p}\left({A}\left({x}\right)+{H}\left({x}\right)\right){is} \\ $$$${divided}\:{by}\:{H}\left({x}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{if}\:{p}\left({x}\right)\neq{c}\:{prove}\:\:{that}\:{p}\left({p}\left({x}\right)\right)−{x}\:{is}\:{divided} \\ $$$${by}\:{p}\left({x}\right)−{x}. \\ $$

Question Number 50362 Answers: 0 Comments: 0

calculate S_1 =Σ_(k=0) ^n C_n ^k S_2 =Σ_(k=0) ^([(n/2)]) C_n ^(2k) S_3 = Σ_(k=0) ^([(n/3)]) C_n ^(3k)

$${calculate}\:{S}_{\mathrm{1}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \\ $$$${S}_{\mathrm{2}} =\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:{C}_{{n}} ^{\mathrm{2}{k}} \\ $$$${S}_{\mathrm{3}} =\:\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{3}}\right]} \:{C}_{{n}} ^{\mathrm{3}{k}} \\ $$

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