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AlgebraQuestion and Answers: Page 341

Question Number 31594 Answers: 1 Comments: 0

Question Number 31591 Answers: 1 Comments: 4

Question Number 31542 Answers: 0 Comments: 0

let p(x)=(1+x+x^2 )^n −(1−x+x^2 )^n ∈C[x] 1) find the roots of p(x) 2) factorize p(x) inside C[x].

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)^{{n}} \:−\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{{n}} \:\in{C}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right]. \\ $$

Question Number 31499 Answers: 0 Comments: 1

find the polynial p wich verify p(x)−p^′ (x)=x^n then calculate ∫_0 ^1 p(x)dx.

$${find}\:{the}\:{polynial}\:{p}\:{wich}\:{verify}\:{p}\left({x}\right)−{p}^{'} \left({x}\right)={x}^{{n}} \:{then} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {p}\left({x}\right){dx}. \\ $$

Question Number 31498 Answers: 0 Comments: 0

find tbe value of Π_(k=1) ^n sin(((kπ)/(n+1))).

$${find}\:{tbe}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{{k}\pi}{{n}+\mathrm{1}}\right). \\ $$

Question Number 31496 Answers: 0 Comments: 1

let a∈]0,π[ and A(x)= x^(2n) −2cos(na)x^n +1 1)factorize inside C[x] A(x) 2) factorize inside R[x] A(x).

$$\left.{let}\:{a}\in\right]\mathrm{0},\pi\left[\:\:\:{and}\:{A}\left({x}\right)=\:{x}^{\mathrm{2}{n}} \:−\mathrm{2}{cos}\left({na}\right){x}^{{n}} \:+\mathrm{1}\right. \\ $$$$\left.\mathrm{1}\right){factorize}\:{inside}\:{C}\left[{x}\right]\:{A}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{A}\left({x}\right). \\ $$

Question Number 31495 Answers: 0 Comments: 0

let give p(x)=(x+j)^n −(x−j)^n with j=e^(i((2π)/3)) 1) find roots of p(x) 2) factorize inside C[ x] p(x) 3)factorize inside R[x] p(x).

$${let}\:{give}\:{p}\left({x}\right)=\left({x}+{j}\right)^{{n}} \:−\left({x}−{j}\right)^{{n}} \:{with}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{C}\left[\:{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){factorize}\:{inside}\:{R}\left[{x}\right]\:{p}\left({x}\right). \\ $$

Question Number 31494 Answers: 0 Comments: 3

prove that x^2 divide (x+1)^n_ −nx−1 .nintegr.

$${prove}\:{that}\:{x}^{\mathrm{2}} \:{divide}\:\left({x}+\mathrm{1}\right)^{\underset{} {{n}}} \:−{nx}−\mathrm{1}\:.{nintegr}. \\ $$

Question Number 31493 Answers: 0 Comments: 0

if (xcosθ +sint)^n =Q(x^2 +1) +R find tbe polynomialR

$${if}\:\left({xcos}\theta\:+{sint}\right)^{{n}} \:={Q}\left({x}^{\mathrm{2}} +\mathrm{1}\right)\:+{R}\:\:{find}\:{tbe}\:{polynomialR} \\ $$

Question Number 31492 Answers: 0 Comments: 0

find all polynomial p(x) wich verify ∀k∈Z ∫_k ^(k+1) p(x)dx=k+1.

$${find}\:{all}\:{polynomial}\:{p}\left({x}\right)\:{wich}\:{verify}\: \\ $$$$\forall{k}\in{Z}\:\:\:\int_{{k}} ^{{k}+\mathrm{1}} {p}\left({x}\right){dx}={k}+\mathrm{1}. \\ $$

Question Number 31491 Answers: 0 Comments: 0

let p(x)= x^n +a_(n−1) x^(n−1) +.... a_1 x +a_o if ξ is roots of p(x) prove that ∣ξ∣ ≤ 1+max_(0≤i≤n−1) ∣a_i ∣

$${let}\:{p}\left({x}\right)=\:{x}^{{n}} \:+{a}_{{n}−\mathrm{1}} {x}^{{n}−\mathrm{1}} \:+....\:{a}_{\mathrm{1}} {x}\:+{a}_{{o}} \\ $$$${if}\:\:\xi\:\:{is}\:{roots}\:{of}\:{p}\left({x}\right)\:{prove}\:{that}\:\mid\xi\mid\:\leqslant\:\mathrm{1}+{max}_{\mathrm{0}\leqslant{i}\leqslant{n}−\mathrm{1}} \:\mid{a}_{{i}} \mid \\ $$

Question Number 31490 Answers: 0 Comments: 0

simplify p(x)= (1+x^2 )(1+x^4 )....(1+x^(2n) ) with n fromN then find the roots of p(x).

$${simplify}\:{p}\left({x}\right)=\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}{n}} \right)\:{with}\:{n}\:{fromN} \\ $$$${then}\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right). \\ $$

Question Number 31485 Answers: 2 Comments: 0

Question Number 31456 Answers: 0 Comments: 2

Find sum of S= (2/3) + (4/3^2 ) + (6/3^3 ) + (8/3^4 ) +......+∞ ?

$$\mathbb{F}{ind}\:{sum}\:{of} \\ $$$${S}=\:\frac{\mathrm{2}}{\mathrm{3}}\:+\:\frac{\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{6}}{\mathrm{3}^{\mathrm{3}} }\:+\:\frac{\mathrm{8}}{\mathrm{3}^{\mathrm{4}} }\:+......+\infty\:? \\ $$

Question Number 31409 Answers: 1 Comments: 0

The maximum area of the triangle whose sides a,b and c satisfy 0≤a≤1 , 1≤b≤2 , 2≤c≤3 is : A) 1 B) 2 C) 1.5 D) 0.5 ?

$${The}\:{maximum}\:{area}\:{of}\:{the}\:{triangle} \\ $$$${whose}\:{sides}\:{a},{b}\:{and}\:{c}\:{satisfy}\: \\ $$$$\mathrm{0}\leqslant{a}\leqslant\mathrm{1}\:,\:\mathrm{1}\leqslant{b}\leqslant\mathrm{2}\:,\:\mathrm{2}\leqslant{c}\leqslant\mathrm{3}\:{is}\:: \\ $$$$\left.{A}\right)\:\mathrm{1} \\ $$$$\left.{B}\right)\:\mathrm{2} \\ $$$$\left.{C}\right)\:\mathrm{1}.\mathrm{5} \\ $$$$\left.{D}\right)\:\mathrm{0}.\mathrm{5}\:\:\:\:\:\:\:? \\ $$

Question Number 31336 Answers: 0 Comments: 1

Find the principal value of z=(1−i)^(1+i) .Hence find the modulus of the result.

$${Find}\:{the}\:{principal}\:{value}\:{of} \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{\mathrm{1}+{i}} .{Hence}\:{find}\:{the} \\ $$$${modulus}\:{of}\:{the}\:{result}. \\ $$

Question Number 31335 Answers: 0 Comments: 3

Find the pricipal value of z=(1−i)^i

$${Find}\:{the}\:{pricipal}\:{value}\:{of}\: \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{{i}} \\ $$

Question Number 31324 Answers: 1 Comments: 0

Question Number 31323 Answers: 1 Comments: 0

Question Number 31320 Answers: 1 Comments: 0

Let p and q are the roots of x^2 − 2mx − 5n = 0 and m and n are the roots of x^2 − 2px − 5q = 0 If p ≠ q ≠ m ≠ n, then the value of p + q + m + n is ...

$$\mathrm{Let}\:{p}\:\mathrm{and}\:{q}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{2}{mx}\:−\:\mathrm{5}{n}\:=\:\mathrm{0} \\ $$$$\mathrm{and}\:{m}\:\mathrm{and}\:{n}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{2}{px}\:−\:\mathrm{5}{q}\:=\:\mathrm{0} \\ $$$$\mathrm{If}\:{p}\:\neq\:{q}\:\neq\:{m}\:\neq\:{n},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${p}\:+\:{q}\:+\:{m}\:+\:{n}\:\mathrm{is}\:... \\ $$

Question Number 31317 Answers: 0 Comments: 1

Question Number 31286 Answers: 1 Comments: 1

Find all set of ordered triple/s (x,y,z), x,y,z∈ℜ, such that x−y=1−z 3(x^2 −y^2 )=5(1−z^2 ) 7(x^3 −y^3 )=19(1−z^3 ). Please show your solution.

$${Find}\:{all}\:{set}\:{of}\:{ordered}\:{triple}/{s}\:\left({x},{y},{z}\right),\:\:{x},{y},{z}\in\Re,\:{such}\:{that} \\ $$$${x}−{y}=\mathrm{1}−{z} \\ $$$$\mathrm{3}\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)=\mathrm{5}\left(\mathrm{1}−{z}^{\mathrm{2}} \right) \\ $$$$\mathrm{7}\left({x}^{\mathrm{3}} −{y}^{\mathrm{3}} \right)=\mathrm{19}\left(\mathrm{1}−{z}^{\mathrm{3}} \right). \\ $$$${Please}\:{show}\:{your}\:{solution}. \\ $$

Question Number 31290 Answers: 1 Comments: 1

Question Number 31259 Answers: 0 Comments: 4

Question Number 31214 Answers: 2 Comments: 2

Question Number 31194 Answers: 2 Comments: 1

Find the remainder when x^(203) −1 is divided by x^4 −1.

$${Find}\:{the}\:{remainder}\:{when}\:{x}^{\mathrm{203}} −\mathrm{1} \\ $$$${is}\:{divided}\:{by}\:{x}^{\mathrm{4}} −\mathrm{1}. \\ $$

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