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AlgebraQuestion and Answers: Page 327

Question Number 46425    Answers: 0   Comments: 2

let p(x)=(x+i)^n −(x−i)^n with i^2 =−1 1) find p(x) at form Σ a_k x^k 2) find the roots of p(x) 3) factorize inside C[x] p(x) 4) factorize inside R[x] the polynom p(x) 5) decompose the fraction F(x)=(1/(p(x)))

$${let}\:{p}\left({x}\right)=\left({x}+{i}\right)^{{n}} −\left({x}−{i}\right)^{{n}} \:\:\:{with}\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{p}\left({x}\right)\:{at}\:{form}\:\Sigma\:{a}_{{k}} {x}^{{k}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{the}\:{polynom}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{5}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{p}\left({x}\right)} \\ $$

Question Number 46420    Answers: 0   Comments: 0

let p(x)=(1+ix)^5 −1 with i^2 =−1 1) solve inside C[x] the equation p(x)=0 2)factorize inside C[x] the polynom p(x)

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{\mathrm{5}} −\mathrm{1}\:\:{with}\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{solve}\:{inside}\:{C}\left[{x}\right]\:{the}\:{equation}\:{p}\left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){factorize}\:{inside}\:{C}\left[{x}\right]\:{the}\:{polynom}\:{p}\left({x}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 46414    Answers: 2   Comments: 1

Question Number 46401    Answers: 0   Comments: 1

Find the sum of the series: (1/(1 + (√x))) , (1/(1 − x)) , (1/(1 − (√x))) , ...

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}:\:\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\sqrt{\mathrm{x}}}\:,\:\:\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{x}}\:,\:\:\frac{\mathrm{1}}{\mathrm{1}\:−\:\sqrt{\mathrm{x}}}\:\:,\:\:...\: \\ $$

Question Number 46395    Answers: 1   Comments: 0

Question Number 46355    Answers: 0   Comments: 0

y/ax−b=j

$${y}/{ax}−{b}={j} \\ $$

Question Number 46348    Answers: 0   Comments: 0

Question Number 46297    Answers: 1   Comments: 0

Find the angle which is equal to one-eighth of its supplement.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{which}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{one}-\mathrm{eighth} \\ $$$$\mathrm{of}\:\mathrm{its}\:\mathrm{supplement}. \\ $$

Question Number 46268    Answers: 2   Comments: 2

please help me! L=lim_(x→1) ((p/(1−x^p ))−(q/(1−x^q ))) , (p,q∈R)

$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}! \\ $$$$\mathrm{L}=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{\mathrm{p}}{\mathrm{1}−\mathrm{x}^{\mathrm{p}} }−\frac{\mathrm{q}}{\mathrm{1}−\mathrm{x}^{\mathrm{q}} }\right)\:,\:\left(\mathrm{p},\mathrm{q}\in\mathbb{R}\right) \\ $$

Question Number 46245    Answers: 1   Comments: 1

(1/(1.3.5))+(1/(3.5.7))+(1/(5.7.9))+.....nth term

$$\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}.\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}.\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{5}.\mathrm{7}.\mathrm{9}}+.....{nth}\:{term} \\ $$

Question Number 46186    Answers: 1   Comments: 1

please help me! S=1^2 q^1 +2^2 q^2 +3^2 q^3 +...+n^2 q^n =?

$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}! \\ $$$$\mathrm{S}=\mathrm{1}^{\mathrm{2}} \mathrm{q}^{\mathrm{1}} +\mathrm{2}^{\mathrm{2}} \mathrm{q}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} \mathrm{q}^{\mathrm{3}} +...+\mathrm{n}^{\mathrm{2}} \mathrm{q}^{\mathrm{n}} =? \\ $$

Question Number 46139    Answers: 2   Comments: 1

A park has the shape of a regular hexagon of sides 2km each. A boy walks a distance of 5km along the sides of the park. What is the direct distance between the start point and the end point?

$$\mathrm{A}\:\mathrm{park}\:\mathrm{has}\:\mathrm{the}\:\mathrm{shape}\:\mathrm{of}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{hexagon} \\ $$$$\mathrm{of}\:\mathrm{sides}\:\mathrm{2km}\:\mathrm{each}.\:\mathrm{A}\:\mathrm{boy}\:\mathrm{walks}\:\mathrm{a}\:\mathrm{distance} \\ $$$$\mathrm{of}\:\mathrm{5km}\:\mathrm{along}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{park}.\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{direct}\:\mathrm{distance}\:\mathrm{between}\: \\ $$$$\mathrm{the}\:\mathrm{start}\:\mathrm{point}\:\mathrm{and}\:\mathrm{the}\:\mathrm{end}\:\mathrm{point}? \\ $$

Question Number 46110    Answers: 1   Comments: 2

Question Number 46088    Answers: 1   Comments: 0

Question Number 45982    Answers: 1   Comments: 1

Find the value(s) of a such that a^x ≥ax with a, x∈R.

$${Find}\:{the}\:{value}\left({s}\right)\:{of}\:{a}\:{such}\:{that} \\ $$$${a}^{{x}} \geqslant{ax}\:{with}\:{a},\:{x}\in{R}. \\ $$

Question Number 45936    Answers: 0   Comments: 1

Question Number 45905    Answers: 2   Comments: 0

Question Number 45592    Answers: 0   Comments: 0

Question Number 45546    Answers: 2   Comments: 1

Simplify: (((√5) + 2))^(1/3) + (((√5) − 2))^(1/3)

$$\mathrm{Simplify}:\:\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}\:\:+\:\:\mathrm{2}}\:\:\:+\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}\:\:−\:\mathrm{2}}\:\: \\ $$

Question Number 45512    Answers: 0   Comments: 11

Calculate: (((2^4 + (1/4)) (4^4 + (1/4))(6^4 + (1/4))(8^4 + (1/4))(10^4 + (1/4))(12^4 + (1/4)))/((1^4 + (1/4))(3^4 + (1/4)) (5^4 + (1/4)) (7^4 + (1/4)) (9^4 + (1/4))(11^4 + (1/4))))

$$\mathrm{Calculate}:\:\:\:\frac{\left(\mathrm{2}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{4}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{6}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{8}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{10}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{12}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)}{\left(\mathrm{1}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{3}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{5}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{7}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{9}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{11}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$

Question Number 45451    Answers: 1   Comments: 0

Question Number 45450    Answers: 1   Comments: 1

Question Number 45422    Answers: 1   Comments: 0

Three consecutive terms of a G.P are the 3rd, 5th and 8th term of an A.P. Find the common ratio.

$$\mathrm{Three}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{a}\:\mathrm{G}.\mathrm{P}\:\mathrm{are}\:\mathrm{the}\:\mathrm{3rd},\:\mathrm{5th}\:\mathrm{and}\:\mathrm{8th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{ratio}. \\ $$

Question Number 45399    Answers: 1   Comments: 0

using your knowlege on Arithmetic progressions, show that A= p(1+(r/(100)))^n

$${using}\:{your}\:{knowlege}\:{on}\:{Arithmetic}\:{progressions}, \\ $$$${show}\:{that}\:\:{A}=\:{p}\left(\mathrm{1}+\frac{{r}}{\mathrm{100}}\right)^{{n}} \\ $$

Question Number 45364    Answers: 1   Comments: 0

Find the sum of n terms: (1/(1.3)) + (1/(3.5)) + ... + (1/((2n − 1)(2n + 1))) = ?

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{terms}:\:\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}}\:+\:...\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:\:=\:? \\ $$

Question Number 45356    Answers: 1   Comments: 0

x=(1/(1+(1/(2+(1/(3+(1/(4+...))))))))=?

$${x}=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{3}+\frac{\mathrm{1}}{\mathrm{4}+...}}}}=? \\ $$

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