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AlgebraQuestion and Answers: Page 307

Question Number 52450 Answers: 0 Comments: 0

let P(x)=(1+ix −x^2 )^n −1 find roots of P(x) and factorize P(x)inside C(x).

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

Question Number 52421 Answers: 0 Comments: 2

Question Number 52412 Answers: 0 Comments: 3

Question Number 52406 Answers: 2 Comments: 0

Find the cube root of 26 + 5(√3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\:\:\:\:\mathrm{26}\:+\:\mathrm{5}\sqrt{\mathrm{3}} \\ $$

Question Number 52397 Answers: 0 Comments: 1

Question Number 52282 Answers: 1 Comments: 7

Find the range of values of x ∣((x − 1)/(2x))∣ ≥ x^2 + 1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{x} \\ $$$$\:\:\:\mid\frac{\mathrm{x}\:−\:\mathrm{1}}{\mathrm{2x}}\mid\:\geqslant\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$

Question Number 52200 Answers: 0 Comments: 0

proof that (√z^2 ) ≠ z , z ∈ C example i=(√(−1)) i^2 =−1 i^4 =1 (√(i^4 )) ≠ i^2 (√1) ≠ −1 1 ≠ −1

$$\mathrm{proof}\:\mathrm{that}\: \\ $$$$\sqrt{{z}^{\mathrm{2}} }\:\neq\:{z}\:,\:{z}\:\in\:\mathbb{C} \\ $$$$\mathrm{example} \\ $$$${i}=\sqrt{−\mathrm{1}} \\ $$$${i}^{\mathrm{2}} =−\mathrm{1} \\ $$$${i}^{\mathrm{4}} =\mathrm{1} \\ $$$$\sqrt{{i}^{\mathrm{4}} \:}\:\neq\:{i}^{\mathrm{2}} \\ $$$$\sqrt{\mathrm{1}}\:\neq\:−\mathrm{1} \\ $$$$\mathrm{1}\:\neq\:−\mathrm{1} \\ $$$$ \\ $$

Question Number 52108 Answers: 1 Comments: 0

Question Number 52087 Answers: 1 Comments: 1

If 1,a_1 ,a_2 ,...,a_(n−1) are n^(th) roots of unity the find the value of (1/(1+1))+(1/(1+a_1 ))+(1/(1+a_2 ))+..+(1/(1+a_(n−1) )) n is odd number

$$\mathrm{If}\:\mathrm{1},{a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,...,{a}_{{n}−\mathrm{1}} \:\mathrm{are}\:{n}^{{th}} \:\mathrm{roots}\:\mathrm{of} \\ $$$$\mathrm{unity}\:\mathrm{the}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{1}+{a}_{\mathrm{1}} }+\frac{\mathrm{1}}{\mathrm{1}+{a}_{\mathrm{2}} }+..+\frac{\mathrm{1}}{\mathrm{1}+{a}_{{n}−\mathrm{1}} } \\ $$$${n}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{number} \\ $$

Question Number 52086 Answers: 1 Comments: 3

If 1,a_(1,) a_2 ,...,a_(n−1) are n^(th) roots of unity, then prove that. (1+a_1 )(1+a_2 )..(1+a_(n−1) )= n if n is even 0 if n is odd

$$\mathrm{If}\:\mathrm{1},{a}_{\mathrm{1},} {a}_{\mathrm{2}} ,...,{a}_{{n}−\mathrm{1}} \:\mathrm{are}\:{n}^{{th}} \:\mathrm{roots}\:\mathrm{of} \\ $$$$\mathrm{unity},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}. \\ $$$$\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)..\left(\mathrm{1}+{a}_{{n}−\mathrm{1}} \right)= \\ $$$${n}\:\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{even} \\ $$$$\mathrm{0}\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{odd} \\ $$

Question Number 52449 Answers: 0 Comments: 0

let j=e^((i2π)/3) and P(x)=(1+jx)^n −(1−jx)^n with n integr natural 1) find roots of P(x) 2)factorize P(x) inside C[x] 3) calculate ∫_0 ^1 P(x)dx. 4) decompose inside C(x) the fraction F(x)=(1/(P(x)))

$${let}\:{j}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:{P}\left({x}\right)=\left(\mathrm{1}+{jx}\right)^{{n}} −\left(\mathrm{1}−{jx}\right)^{{n}} \:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {P}\left({x}\right){dx}. \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{P}\left({x}\right)} \\ $$

Question Number 52007 Answers: 2 Comments: 0

Solve: ((x/4))^(log_5 50x) = x^6

$$\mathrm{Solve}:\:\:\:\:\:\:\:\:\:\:\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{4}}\right)^{\boldsymbol{\mathrm{log}}_{\mathrm{5}} \mathrm{50}\boldsymbol{\mathrm{x}}} \:\:\:=\:\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{6}} \\ $$

Question Number 51922 Answers: 1 Comments: 0

find the value of... 1−(1/(1+(1/(i/(1+(i/(1+i))))))) pls help.

$${find}\:{the}\:{value}\:{of}... \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\frac{{i}}{\mathrm{1}+\frac{{i}}{\mathrm{1}+{i}}}}} \\ $$$${pls}\:{help}. \\ $$

Question Number 51843 Answers: 1 Comments: 3

If ax^2 +bx+c+i=0 has purely imaginary roots where a,b,c are non−zero real. answer given: a=b^2 c I think question is wrong since if z_1 and z_2 are roots than z_1 +z_2 =−(b/a) purely imaginary=purely real not possible Can some point a mistake.

$${If}\:{ax}^{\mathrm{2}} +{bx}+{c}+{i}=\mathrm{0}\:\mathrm{has}\:\mathrm{purely} \\ $$$$\mathrm{imaginary}\:\mathrm{roots}\:\mathrm{where}\: \\ $$$${a},{b},{c}\:{are}\:{non}−{zero}\:{real}. \\ $$$${answer}\:{given}:\:{a}={b}^{\mathrm{2}} {c} \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{question}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{since}\:\mathrm{if}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{roots}\:\mathrm{than} \\ $$$${z}_{\mathrm{1}} +{z}_{\mathrm{2}} =−\frac{{b}}{{a}} \\ $$$${purely}\:{imaginary}={purely}\:{real} \\ $$$${not}\:{possible} \\ $$$$\mathrm{Can}\:\mathrm{some}\:\mathrm{point}\:\mathrm{a}\:\mathrm{mistake}. \\ $$

Question Number 51772 Answers: 2 Comments: 0

x^3 +12x+12=0 Express your answer in surd form

$$\mathrm{x}^{\mathrm{3}} +\mathrm{12x}+\mathrm{12}=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Express}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{surd}\:\mathrm{form} \\ $$$$ \\ $$

Question Number 51691 Answers: 1 Comments: 0