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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)

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

Question Number 50359    Answers: 0   Comments: 0

1) calculate Σ_(n=0) ^∞ (−1)^k C_n ^k (1/(k+1)) 2)calculate S_n (p)=Σ_(k=0) ^n (−1)^k (C_n ^k /(p+k+1)) p integr natural.

$$\left.\mathrm{1}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:\frac{\mathrm{1}}{{k}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{S}_{{n}} \left({p}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:\:\:\frac{{C}_{{n}} ^{{k}} }{{p}+{k}+\mathrm{1}} \\ $$$${p}\:{integr}\:{natural}. \\ $$

Question Number 50356    Answers: 0   Comments: 0

devompose inside C[x] and R[x] the polynom 1)x^4 +1 2)x^6 −1 3)x^8 +x^4 +1

$${devompose}\:{inside}\:{C}\left[{x}\right]\:{and}\:{R}\left[{x}\right]\:{the}\:{polynom} \\ $$$$\left.\mathrm{1}\right){x}^{\mathrm{4}} +\mathrm{1}\:\:\:\: \\ $$$$\left.\mathrm{2}\right){x}^{\mathrm{6}} −\mathrm{1} \\ $$$$\left.\mathrm{3}\right){x}^{\mathrm{8}} \:+{x}^{\mathrm{4}} \:+\mathrm{1} \\ $$

Question Number 50355    Answers: 0   Comments: 0

let (α_k ) (k∈[[0,n−1]] the n^(eme) roots of 1 calculate p(x,y)=(x+α_0 y)(x+α_1 y)....(x+α_(n−1) y)

$${let}\:\left(\alpha_{{k}} \right)\:\:\left({k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:{the}\:{n}^{{eme}} \:{roots}\:{of}\:\mathrm{1}\right. \\ $$$${calculate}\:{p}\left({x},{y}\right)=\left({x}+\alpha_{\mathrm{0}} {y}\right)\left({x}+\alpha_{\mathrm{1}} {y}\right)....\left({x}+\alpha_{{n}−\mathrm{1}} {y}\right) \\ $$

Question Number 50354    Answers: 0   Comments: 0

((x)=a_k ) _(1≤k≤n) is a sequence of reals let p(x) =Π_(k=1) ^n (cos(a_k )+xsin(a_k )) if p(x)=(x^2 +1)q +r find q and r

$$\left(\left({x}\right)={a}_{{k}} \right)\:_{\mathrm{1}\leqslant{k}\leqslant{n}} \:{is}\:{a}\:{sequence}\:{of}\:{reals}\:{let} \\ $$$${p}\left({x}\right)\:=\prod_{{k}=\mathrm{1}} ^{{n}} \left({cos}\left({a}_{{k}} \right)+{xsin}\left({a}_{{k}} \right)\right) \\ $$$${if}\:{p}\left({x}\right)=\left({x}^{\mathrm{2}} +\mathrm{1}\right){q}\:+{r}\:\:\:{find}\:{q}\:{and}\:{r} \\ $$$$ \\ $$

Question Number 50353    Answers: 0   Comments: 0

let p(x)=x^(4n) −x^(3n) +x^(2n) −x^n +1 and q(x)=x^4 −x^3 +x^2 −x+1 determine the integr n to have q divide p.

$${let}\:{p}\left({x}\right)={x}^{\mathrm{4}{n}} −{x}^{\mathrm{3}{n}} +{x}^{\mathrm{2}{n}} −{x}^{{n}} +\mathrm{1}\:{and} \\ $$$${q}\left({x}\right)={x}^{\mathrm{4}} −{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −{x}+\mathrm{1}\:\:{determine}\:{the}\:{integr}\:{n} \\ $$$${to}\:{have}\:{q}\:{divide}\:{p}. \\ $$

Question Number 50328    Answers: 2   Comments: 0

Question Number 50279    Answers: 2   Comments: 0

{ ((x+y=6)),((y+z=10)) :} (x,y,z>0)

$$\begin{cases}{\mathrm{x}+\mathrm{y}=\mathrm{6}}\\{\mathrm{y}+\mathrm{z}=\mathrm{10}}\end{cases}\:\:\left(\mathrm{x},\mathrm{y},\mathrm{z}>\mathrm{0}\right) \\ $$

Question Number 50278    Answers: 1   Comments: 4

(√(a+(√(a−x)))) + (√(a−(√(a+x)))) = 2x please i beg u guys please solve this question

$$\sqrt{\mathrm{a}+\sqrt{\mathrm{a}−\mathrm{x}}}\:+\:\sqrt{\mathrm{a}−\sqrt{\mathrm{a}+\mathrm{x}}}\:=\:\mathrm{2x} \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{beg}\:\mathrm{u}\:\mathrm{guys}\: \\ $$$$\mathrm{please}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{question} \\ $$

Question Number 50226    Answers: 0   Comments: 1

Three prizes are awarded each for getting more than 80%marks, 98% attendance and good behaviour in the college.In how many ways the prozes can be awarded if 15 student of the college are eligible for the three prizes?

$$\mathrm{Three}\:\mathrm{prizes}\:\mathrm{are}\:\mathrm{awarded}\:\mathrm{each}\:\mathrm{for} \\ $$$$\mathrm{getting}\:\mathrm{more}\:\mathrm{than}\:\mathrm{80\%marks}, \\ $$$$\mathrm{98\%}\:\mathrm{attendance}\:\mathrm{and}\:\mathrm{good} \\ $$$$\mathrm{behaviour}\:\mathrm{in}\:\mathrm{the}\:\mathrm{college}.\mathrm{In}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{ways}\:\mathrm{the}\:\mathrm{prozes}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{awarded}\:\mathrm{if}\:\mathrm{15}\:\mathrm{student}\:\mathrm{of}\:\mathrm{the}\:\mathrm{college} \\ $$$$\mathrm{are}\:\mathrm{eligible}\:\mathrm{for}\:\mathrm{the}\:\mathrm{three}\:\mathrm{prizes}? \\ $$

Question Number 50349    Answers: 0   Comments: 0

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