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AlgebraQuestion and Answers: Page 346

Question Number 26999 Answers: 1 Comments: 0

calculate Π_(k=1) ^n cos((a/2^k )) and0<a<π then find the value of lim_(n−>∝) Σ_(k=1) ^n ln(cos((a/2^k ))).

$${calculate}\:\prod_{{k}=\mathrm{1}} ^{{n}} {cos}\left(\frac{{a}}{\mathrm{2}^{{k}} }\right)\:\:{and}\mathrm{0}<{a}<\pi\:\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$${lim}_{{n}−>\propto} \:\sum_{{k}=\mathrm{1}} ^{{n}} {ln}\left({cos}\left(\frac{{a}}{\mathrm{2}^{{k}} }\right)\right). \\ $$

Question Number 26998 Answers: 0 Comments: 0

smlify X= Π_(p=2) ^n ((p^3 −1)/(p^3 +1)) by using 1,j,j^2 and j=e^((i2π)/3) .

$${smlify}\:{X}=\:\:\prod_{{p}=\mathrm{2}} ^{{n}} \frac{{p}^{\mathrm{3}} −\mathrm{1}}{{p}^{\mathrm{3}} \:+\mathrm{1}}\:{by}\:{using}\:\mathrm{1},{j},{j}^{\mathrm{2}} {and}\:{j}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} . \\ $$

Question Number 26997 Answers: 0 Comments: 3

let give ξ ∈C and ξ^n =1 (ξ is the n^(me) root of 1) simplify A= 1+ξ^p +ξ^(2p) +... +ξ^((n−1)p) and B= 1+2ξ +3ξ^2 +...+nξ^(n−1) .

$${let}\:{give}\:\xi\:\in\mathbb{C}\:{and}\:\xi^{{n}} =\mathrm{1}\:\left(\xi\:{is}\:{the}\:{n}^{{me}} \:{root}\:{of}\:\mathrm{1}\right) \\ $$$${simplify}\:\:{A}=\:\mathrm{1}+\xi^{{p}} +\xi^{\mathrm{2}{p}} +...\:+\xi^{\left({n}−\mathrm{1}\right){p}} \\ $$$${and}\:{B}=\:\mathrm{1}+\mathrm{2}\xi\:+\mathrm{3}\xi^{\mathrm{2}} +...+{n}\xi^{{n}−\mathrm{1}} . \\ $$

Question Number 26942 Answers: 1 Comments: 0

Question Number 27000 Answers: 0 Comments: 1

P is a polynomial havng n roots (x_i )_(1≤i≤n) with x_i ≠ x_j for i≠ j find the values of Σ_(k1) ^(k=n) (1/(x−x_k )) and Σ_(k=1) ^n (1/((x−x_k )^2 )) .

$${P}\:{is}\:{a}\:{polynomial}\:{havng}\:{n}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:\:{with}\:{x}_{{i}} \neq\:{x}_{{j}} \:{for}\:{i}\neq\:{j} \\ $$$${find}\:{the}\:{values}\:{of}\:\sum_{{k}\mathrm{1}} ^{{k}={n}} \frac{\mathrm{1}}{{x}−{x}_{{k}} }\:\:{and}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\left({x}−{x}_{{k}} \right)^{\mathrm{2}} }\:. \\ $$

Question Number 26733 Answers: 1 Comments: 1

Question Number 27002 Answers: 2 Comments: 4

Question Number 26694 Answers: 1 Comments: 0

divide x^6 −y^6 by the product of x^2 +x^ y+y^(2 ) and x−y.

$${divide}\:{x}^{\mathrm{6}} −{y}^{\mathrm{6}} \:{by}\:{the}\:{product}\:{of}\:{x}^{\mathrm{2}} +{x}^{} {y}+{y}^{\mathrm{2}\:} \:{and}\:{x}−{y}. \\ $$

Question Number 26680 Answers: 1 Comments: 0

(b−c)x^2 +(c−a)x+(a−b)=0 if the eqation roots are eqal you proved that 2b=a+c.

$$\left({b}−{c}\right){x}^{\mathrm{2}} +\left({c}−{a}\right){x}+\left({a}−{b}\right)=\mathrm{0}\:{if}\:{the}\: \\ $$$${eqation}\:{roots}\:\:{are}\:{eqal}\:{you}\:{proved}\:{that} \\ $$$$\mathrm{2}{b}={a}+{c}. \\ $$

Question Number 26642 Answers: 2 Comments: 0

9x^2 +41x−204=0. solved it.

$$\mathrm{9}{x}^{\mathrm{2}} +\mathrm{41}{x}−\mathrm{204}=\mathrm{0}.\:{solved}\:{it}. \\ $$

Question Number 26638 Answers: 1 Comments: 0

x^3 −8x^2 +7 factorise it.

$$\mathrm{x}^{\mathrm{3}} −\mathrm{8x}^{\mathrm{2}} +\mathrm{7}\:\:\mathrm{factorise}\:\mathrm{it}. \\ $$

Question Number 26583 Answers: 0 Comments: 2

find the decomposition in C[x] then R[x] for the rationsl fraction F(x)= ((1 )/(x^(2n) −1)) .with n integer not 0

$${find}\:{the}\:{decomposition}\:{in}\:\mathbb{C}\left[{x}\right]\:{then}\:\mathbb{R}\left[{x}\right] \\ $$$${for}\:{the}\:{rationsl}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}\:}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:\:.{with}\:{n}\:{integer}\:{not}\:\mathrm{0} \\ $$

Question Number 26582 Answers: 0 Comments: 1

p is a polynomial having the roots x_1 ,x_2 ,...x_n with x_i ≠ x_j fori≠j give the decomposition of the fravtion F(x)= ((p^′ (x))/(p(x)))

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{the}\:{roots}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...{x}_{{n}} \\ $$$${with}\:{x}_{{i}} \neq\:{x}_{{j}} \:{fori}\neq{j}\:{give}\:{the}\:{decomposition} \\ $$$${of}\:{the}\:{fravtion}\:{F}\left({x}\right)=\:\frac{{p}^{'} \left({x}\right)}{{p}\left({x}\right)} \\ $$

Question Number 29176 Answers: 3 Comments: 1

∫sec x dx=?

$$\int{sec}\:{x}\:{dx}=? \\ $$

Question Number 26461 Answers: 0 Comments: 1

Question Number 26405 Answers: 1 Comments: 0

Question Number 26348 Answers: 1 Comments: 1

Question Number 26347 Answers: 0 Comments: 0

solve: 5x(1 + (1/(x^2 + y^2 ))) = 12 ..... equation (i) 5y(1 − (1/(x^2 + y^2 ))) = 4 ...... equation (ii)

$$\mathrm{solve}: \\ $$$$\mathrm{5x}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\right)\:=\:\mathrm{12}\:\:\:\:\:\:\:\:.....\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{5y}\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\right)\:=\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:......\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 26274 Answers: 0 Comments: 2

could there be an analytical or numerical meghod for solving this non-linear simultaneous equation x+y=5 x^x +y^y =31 please help if possible

$${could}\:{there}\:{be}\:{an}\:{analytical}\:{or} \\ $$$${numerical}\:{meghod}\:{for}\:{solving} \\ $$$${this}\:{non}-{linear}\:{simultaneous} \\ $$$${equation} \\ $$$${x}+{y}=\mathrm{5} \\ $$$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$$ \\ $$$${please}\:{help}\:{if}\:{possible} \\ $$

Question Number 26269 Answers: 1 Comments: 0

if x^x =2 what is the value of x?

$${if}\:{x}^{{x}} =\mathrm{2}\:{what}\:{is}\:{the}\:{value}\:{of}\:{x}? \\ $$

Question Number 26250 Answers: 1 Comments: 0

someone should help witb solution please x^3 +y^3 =3x^2 −6x−3y+4 x^2 −y^2 −6x+y−10=(√((y+5)))−(√((4x+y)))

$${someone}\:{should}\:{help}\:{witb}\:{solution}\:{please} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{3}{x}^{\mathrm{2}} −\mathrm{6}{x}−\mathrm{3}{y}+\mathrm{4} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} −\mathrm{6}{x}+{y}−\mathrm{10}=\sqrt{\left({y}+\mathrm{5}\right)}−\sqrt{\left(\mathrm{4}{x}+{y}\right)} \\ $$

Question Number 26235 Answers: 1 Comments: 0

Find the real root of x^2 +(1/x)=c .

$${Find}\:{the}\:{real}\:{root}\:{of} \\ $$$$\:{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}}={c}\:. \\ $$

Question Number 26229 Answers: 1 Comments: 1

after factorise (x2+2x+1)we get

$$\mathrm{after}\:\mathrm{factorise}\:\left(\mathrm{x2}+\mathrm{2x}+\mathrm{1}\right)\mathrm{we}\:\mathrm{get} \\ $$

Question Number 26228 Answers: 0 Comments: 1

if x^4 +(1/x^4 )=322 find x^3 −(1/x^3 )

$$\mathrm{if}\:\mathrm{x}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }=\mathrm{322}\:\mathrm{find}\:\mathrm{x}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 26227 Answers: 1 Comments: 0

if x^2 +(1/x^2 )=98 find x^3 +(1/x^3 )

$$\mathrm{if}\:\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }=\mathrm{98}\:\mathrm{find}\:\mathrm{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 26224 Answers: 1 Comments: 1

find the value of x−(1/x).when x^4 +(1/x^4 )=332

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}.\mathrm{when}\:{x}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }=\mathrm{332} \\ $$

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