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AlgebraQuestion and Answers: Page 300

Question Number 64186    Answers: 1   Comments: 0

Question Number 64185    Answers: 2   Comments: 3

Question Number 64174    Answers: 0   Comments: 0

2^n / 5^2^n + 1 infinite series sum ffom 0 to infinity

$$\mathrm{2}^{{n}} \:/\:\mathrm{5}^{\mathrm{2}^{{n}} } \:+\:\mathrm{1}\:{infinite}\:{series}\:{sum}\:{ffom}\:\mathrm{0}\:{to}\: \\ $$$${infinity} \\ $$

Question Number 64130    Answers: 2   Comments: 1

(√(4x+((12)/x)))=((x^2 +7)/(x+1)) x=?

$$\sqrt{\mathrm{4}\boldsymbol{\mathrm{x}}+\frac{\mathrm{12}}{\boldsymbol{\mathrm{x}}}}=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{7}}{\boldsymbol{\mathrm{x}}+\mathrm{1}} \\ $$$$\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 64126    Answers: 1   Comments: 0

Question Number 64111    Answers: 0   Comments: 0

Question Number 64112    Answers: 2   Comments: 1

Question Number 64066    Answers: 1   Comments: 0

let α ,β and λ the roots of x^3 +2x−1 =0 find the value of A =α^2 +β^2 +λ^2 and B =α^3 +β^3 +λ^3 .

$${let}\:\alpha\:,\beta\:{and}\:\lambda\:{the}\:{roots}\:{of}\:{x}^{\mathrm{3}} +\mathrm{2}{x}−\mathrm{1}\:=\mathrm{0}\:{find}\:{the}\:{value}\:{of} \\ $$$${A}\:=\alpha^{\mathrm{2}} \:+\beta^{\mathrm{2}} \:+\lambda^{\mathrm{2}} \:{and}\:\:{B}\:=\alpha^{\mathrm{3}} \:+\beta^{\mathrm{3}} \:+\lambda^{\mathrm{3}} \:. \\ $$

Question Number 64061    Answers: 1   Comments: 0

(2x+3)^2 +25/(x+3)^2 =(√2)

$$\left(\mathrm{2}{x}+\mathrm{3}\right)^{\mathrm{2}} +\mathrm{25}/\left({x}+\mathrm{3}\right)^{\mathrm{2}} =\sqrt{\mathrm{2}} \\ $$

Question Number 63958    Answers: 0   Comments: 1

x^6 −3x^5 +4x^4 −6x^3 +5x^2 −3x+2=0

$${x}^{\mathrm{6}} −\mathrm{3}{x}^{\mathrm{5}} +\mathrm{4}{x}^{\mathrm{4}} −\mathrm{6}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}=\mathrm{0} \\ $$

Question Number 63945    Answers: 0   Comments: 1

solve at Z^2 x^2 −2y^2 +xy +2 =0

$${solve}\:{at}\:{Z}^{\mathrm{2}} \:\:{x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \:+{xy}\:+\mathrm{2}\:=\mathrm{0} \\ $$

Question Number 63930    Answers: 0   Comments: 0

Question Number 63893    Answers: 0   Comments: 1

1) simplify W_n (z)=(1+z)(1+z^2 )....(1+z^2^n ) (z from C) 2) simplify P_n (θ) =(1+e^(iθ) )(1+e^(2iθ) ).....(1+e^(i2^n θ) ) and sove P_n (θ)=0

$$\left.\mathrm{1}\right)\:{simplify}\:{W}_{{n}} \left({z}\right)=\left(\mathrm{1}+{z}\right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)....\left(\mathrm{1}+{z}^{\mathrm{2}^{{n}} } \right)\:\left({z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{2}\right)\:{simplify}\:{P}_{{n}} \left(\theta\right)\:=\left(\mathrm{1}+{e}^{{i}\theta} \right)\left(\mathrm{1}+{e}^{\mathrm{2}{i}\theta} \right).....\left(\mathrm{1}+{e}^{{i}\mathrm{2}^{{n}} \theta} \right)\:{and}\:{sove} \\ $$$${P}_{{n}} \left(\theta\right)=\mathrm{0} \\ $$

Question Number 63858    Answers: 1   Comments: 0

if a_1 , a_2 , a_3 , a_4 are the coefficient of any four four consecutive terms in the expansion of (1+x)^n then (a_1 /(a_2 +a_1 ))+(a_3 /(a_3 +a_4 )) is equal to...

$$\mathrm{if}\:\mathrm{a}_{\mathrm{1}} ,\:\mathrm{a}_{\mathrm{2}} ,\:\mathrm{a}_{\mathrm{3}} ,\:\mathrm{a}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{any}\:\mathrm{four}\:\mathrm{four}\:\mathrm{consecutive} \\ $$$$\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} \\ $$$$\mathrm{then}\:\frac{\mathrm{a}_{\mathrm{1}} }{\mathrm{a}_{\mathrm{2}} +\mathrm{a}_{\mathrm{1}} }+\frac{\mathrm{a}_{\mathrm{3}} }{\mathrm{a}_{\mathrm{3}} +\mathrm{a}_{\mathrm{4}} }\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}... \\ $$

Question Number 63803    Answers: 1   Comments: 4

Question Number 63784    Answers: 0   Comments: 6

question 63639 again prove: ∀z∈C: ∣z+1∣+∣z^2 +z+1∣+∣z^3 +1∣≥1

$$\mathrm{question}\:\mathrm{63639}\:\mathrm{again} \\ $$$$\mathrm{prove}: \\ $$$$\forall{z}\in\mathbb{C}:\:\mid{z}+\mathrm{1}\mid+\mid{z}^{\mathrm{2}} +{z}+\mathrm{1}\mid+\mid{z}^{\mathrm{3}} +\mathrm{1}\mid\geqslant\mathrm{1} \\ $$

Question Number 63678    Answers: 0   Comments: 1

Question Number 63642    Answers: 2   Comments: 1

Question Number 63639    Answers: 0   Comments: 0

Question Number 63602    Answers: 2   Comments: 0

32x^3 −48x^2 −22x−3=0

$$\mathrm{32}{x}^{\mathrm{3}} −\mathrm{48}{x}^{\mathrm{2}} −\mathrm{22}{x}−\mathrm{3}=\mathrm{0} \\ $$

Question Number 63574    Answers: 0   Comments: 12

prove that Σ_(k = 1) ^∞ (1/(k(2k + 1))) = 2 − 2ln(2)

$$\mathrm{prove}\:\mathrm{that}\:\:\:\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{k}\left(\mathrm{2k}\:+\:\mathrm{1}\right)}\:\:=\:\:\mathrm{2}\:−\:\mathrm{2ln}\left(\mathrm{2}\right) \\ $$

Question Number 63573    Answers: 0   Comments: 0

Question Number 63565    Answers: 0   Comments: 0

Question Number 63522    Answers: 0   Comments: 2

Question Number 63485    Answers: 1   Comments: 0

f(x−3)+f(x)=2x−3 F(2)=0. F(−2)=?

$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}−\mathrm{3}\right)+\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{2}\boldsymbol{\mathrm{x}}−\mathrm{3} \\ $$$$\boldsymbol{\mathrm{F}}\left(\mathrm{2}\right)=\mathrm{0}. \\ $$$$\boldsymbol{\mathrm{F}}\left(−\mathrm{2}\right)=? \\ $$

Question Number 63474    Answers: 1   Comments: 0

let P(x)=x^2 +(1/2)x+b and Q(x)=x^2 +cx+d be to polynomials with real coefficient such that P(x) Q(x)=Q(P(x)) find all the real roots of P(Q(x))=0

$${let}\:{P}\left({x}\right)={x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{x}+{b} \\ $$$$ \\ $$$${and}\:{Q}\left({x}\right)={x}^{\mathrm{2}} +{cx}+{d} \\ $$$$ \\ $$$${be}\:{to}\:{polynomials}\:{with}\:{real}\:{coefficient}\:{such}\:{that} \\ $$$$ \\ $$$${P}\left({x}\right)\:{Q}\left({x}\right)={Q}\left({P}\left({x}\right)\right) \\ $$$$ \\ $$$${find}\:{all}\:{the}\:{real}\:{roots}\:{of}\:{P}\left({Q}\left({x}\right)\right)=\mathrm{0} \\ $$

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