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AlgebraQuestion and Answers: Page 336

Question Number 34669 Answers: 0 Comments: 1

let P(x)=(1+x+ix^2 )^n −(1+x −ix^2 )^n 1) find the roots of P(x) 2) factorize inside C[x] P(x) 3) factorize indide R[x] P(x).

$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{x}+{ix}^{\mathrm{2}} \right)^{{n}} \:−\left(\mathrm{1}+{x}\:−{ix}^{\mathrm{2}} \right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{P}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{indide}\:{R}\left[{x}\right]\:{P}\left({x}\right). \\ $$

Question Number 34668 Answers: 0 Comments: 0

find the roots of?p(x) = x^(2n) −2x^n cos(nθ) +1 2)?factorize p(x)

$${find}\:{the}\:{roots}\:{of}?{p}\left({x}\right)\:=\:{x}^{\mathrm{2}{n}} \:−\mathrm{2}{x}^{{n}} \:{cos}\left({n}\theta\right)\:+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)?{factorize}\:{p}\left({x}\right)\: \\ $$

Question Number 34667 Answers: 0 Comments: 0

solve (x+1)^n = e^(2ina) then find the value of P_n = Π_(k=0) ^(n−1) sin(a +((kπ)/n))

$${solve}\:\:\left({x}+\mathrm{1}\right)^{{n}} \:=\:{e}^{\mathrm{2}{ina}} \:\:\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$${P}_{{n}} =\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left({a}\:+\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 34653 Answers: 0 Comments: 0

Question Number 34615 Answers: 0 Comments: 0

decompose inside R(x) the fraction F(x)= (x^2 /((x+1)^5 ( x+3)^8 ))

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}+\mathrm{1}\right)^{\mathrm{5}} \left(\:{x}+\mathrm{3}\right)^{\mathrm{8}} } \\ $$

Question Number 34607 Answers: 0 Comments: 0

let p∈C[x] degp=n (x_i )_(1≤k≤n) the roots of p(x) a∈C?/p(a)≠0 1) calculate S_1 = Σ_(k=1) ^n (1/(x_k −a)) interms of p,p^′ and a 2)calculste S_2 =Σ_(k=1) ^n (1/((x_k −a)^2 )) interms of p,p^, p^(′′) and a.

$${let}\:{p}\in{C}\left[{x}\right]\:{degp}={n}\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} {the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$${a}\in{C}?/{p}\left({a}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{\mathrm{1}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}_{{k}} −{a}}\:{interms}\:{of}\:{p},{p}^{'} \:{and}\:{a} \\ $$$$\left.\mathrm{2}\right){calculste}\:{S}_{\mathrm{2}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({x}_{{k}} −{a}\right)^{\mathrm{2}} }\:\:{interms}\:{of}\:{p},{p}^{,} \\ $$$${p}^{''} \:{and}\:{a}. \\ $$

Question Number 34606 Answers: 0 Comments: 0

let give p(x)=(x+1)^n −(x−1)^n 1) factorize p(x) inside C[x] 2) find the value of Π_(k=1) ^p cotan(((kπ)/(2p+1)))

$${let}\:{give}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} \:−\left({x}−\mathrm{1}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:\:{cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right) \\ $$

Question Number 34604 Answers: 0 Comments: 0

let p(x)= x^n +x+1 ∈C[x] and z∈C/p(z)=0 prove that ∣z∣<2 .

$${let}\:\:{p}\left({x}\right)=\:{x}^{{n}} \:+{x}+\mathrm{1}\:\in{C}\left[{x}\right]\:{and}\:{z}\in{C}/{p}\left({z}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:\mid{z}\mid<\mathrm{2}\:. \\ $$

Question Number 34603 Answers: 0 Comments: 0

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

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

Question Number 34602 Answers: 0 Comments: 0

simplify Σ_(k=0) ^n ((k/n) −α)^2 C_n ^k x^k (1−x)^(n−k ) α∈C.

$${simplify}\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}\:−\alpha\right)^{\mathrm{2}} {C}_{{n}} ^{{k}} \:{x}^{{k}} \left(\mathrm{1}−{x}\right)^{{n}−{k}\:} \\ $$$$\alpha\in{C}. \\ $$

Question Number 34595 Answers: 0 Comments: 0

simplify Σ_(k=1) ^n (((−1)^(k−1) )/k) C_n ^k

$${simplify}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{C}_{{n}} ^{{k}} \\ $$

Question Number 34614 Answers: 0 Comments: 1

decompose inside R(x) the fraction F(x)= (1/((x−3)^6 (x+2))) .

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\frac{\mathrm{1}}{\left({x}−\mathrm{3}\right)^{\mathrm{6}} \left({x}+\mathrm{2}\right)}\:. \\ $$

Question Number 34533 Answers: 2 Comments: 5

Solve for x : 5 log_4 x + 48 log_x 4 = (x/8)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\::\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\mathrm{log}_{\mathrm{4}} \mathrm{x}\:\:\:+\:\:\:\mathrm{48}\:\mathrm{log}_{\mathrm{x}} \mathrm{4}\:\:\:\:=\:\:\:\:\frac{\mathrm{x}}{\mathrm{8}} \\ $$

Question Number 34429 Answers: 2 Comments: 0

Prove that 3^m +3^n +1 is not a perfect square. where m and n are positive integers.

$${Prove}\:{that} \\ $$$$\mathrm{3}^{{m}} +\mathrm{3}^{{n}} +\mathrm{1}\:{is}\:{not}\:{a}\:{perfect}\:{square}. \\ $$$${where}\:{m}\:{and}\:{n}\:{are}\:{positive}\:{integers}. \\ $$

Question Number 34422 Answers: 2 Comments: 0

Show that ((1+x)/(1+(√(1+x)) )) +((1−x)/(1−(√(1−x )))) =1 when x=((√(3 ))/2)

$${Show}\:{that} \\ $$$$\frac{\mathrm{1}+{x}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}}\:}\:+\frac{\mathrm{1}−{x}}{\mathrm{1}−\sqrt{\mathrm{1}−{x}\:}}\:=\mathrm{1}\:{when}\:{x}=\frac{\sqrt{\mathrm{3}\:}}{\mathrm{2}} \\ $$

Question Number 34357 Answers: 1 Comments: 0

If a+b+c=0 prove that i)((a/(b+c))+ (b/(c+a)) +(c/(a+b)))(((b+c)/a) +((c+a)/b) +((a+b)/c))=9 ii)((a/(b−c)) +(b/(c−a)) +(c/(a−b)))(((b−c)/a) +((c−a)/b) +((a−b)/c))=9

$${If}\:{a}+{b}+{c}=\mathrm{0}\:\:{prove}\:{that} \\ $$$$\left.{i}\right)\left(\frac{{a}}{{b}+{c}}+\:\frac{{b}}{{c}+{a}}\:+\frac{{c}}{{a}+{b}}\right)\left(\frac{{b}+{c}}{{a}}\:+\frac{{c}+{a}}{{b}}\:+\frac{{a}+{b}}{{c}}\right)=\mathrm{9} \\ $$$$\left.{ii}\right)\left(\frac{{a}}{{b}−{c}}\:+\frac{{b}}{{c}−{a}}\:+\frac{{c}}{{a}−{b}}\right)\left(\frac{{b}−{c}}{{a}}\:+\frac{{c}−{a}}{{b}}\:+\frac{{a}−{b}}{{c}}\right)=\mathrm{9} \\ $$

Question Number 34307 Answers: 0 Comments: 1

let give A_n = (((1 (α/n))),((−(α/n) 1)) ) calculate lim_(n→+∞) A_n ^n .

$${let}\:{give}\:{A}_{{n}} =\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\frac{\alpha}{{n}}}\\{−\frac{\alpha}{{n}}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ^{{n}} \:\:\:\:. \\ $$

Question Number 34305 Answers: 0 Comments: 1

let] A = (((1 1 0)),((1 1 1)) ) (0 1 1 1)find the caractetistic polynom of A 2) calculate A^n

$$\left.{let}\right]\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right. \\ $$$$\left.\mathrm{1}\right){find}\:{the}\:{caractetistic}\:{polynom}\:{of}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \\ $$

Question Number 34215 Answers: 0 Comments: 0

find the polynome p_n wich verify p_n (0)=0 and ∀ x ∈ R p_n (x)−p_n (x−1) =x^n

$${find}\:{the}\:{polynome}\:{p}_{{n}} \:{wich}\:{verify}\:{p}_{{n}} \left(\mathrm{0}\right)=\mathrm{0}\:{and} \\ $$$$\forall\:{x}\:\in\:{R}\:\:{p}_{{n}} \left({x}\right)−{p}_{{n}} \left({x}−\mathrm{1}\right)\:={x}^{{n}} \\ $$

Question Number 34211 Answers: 1 Comments: 0

let x and y such that 2x^2 +4x−2y=0 y^2 −(x+6)^2 =0 find the possibles value of x+y