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AlgebraQuestion and Answers: Page 322

Question Number 39636 Answers: 1 Comments: 0

let P_α (x) =x^3 +2α x −3 1) determine the roots of P_α 2) determine the roots of P_(−1)

$l e t P_{α} (x) = x^{3} + 2 α x - 3$ $1) d e t e r m i n e t h e r o o t s o f P_{α}$ $2) d e t e r m i n e t h e r o o t s o f P_{- 1}$

Question Number 39524 Answers: 0 Comments: 0

Question Number 39332 Answers: 0 Comments: 1

1) simplify S_n (x)=Σ_(k=1) ^n sin^2 (kx) 2)simplify A_n =Σ_(k=1) ^n sin^2 (((kπ)/n))

$1) s i m p l i f y S_{n} (x) = \sum_{k = 1}^{n} {s i n}^{2} (k x)$ $2) s i m p l i f y A_{n} = \sum_{k = 1}^{n} {s i n}^{2} (\frac{k π}{n})$

Question Number 39218 Answers: 1 Comments: 0

Question Number 39026 Answers: 2 Comments: 0

find the roots of 8x^3 −4x−1 =0

$f i n d t h e r o o t s o f 8 x^{3} - 4 x - 1 = 0$

Question Number 39022 Answers: 0 Comments: 1

let p(x)= (1+e^(iθ) x)^n −(1−e^(iθ) x)^n with n integr natural 1) find the roots of p(x) 2) fctorize inside C[x] p(x) 3) factorize inside R[x] p(x). θ ∈R

$l e t p (x) = {(1 + e^{i θ} x)}^{n} - {(1 - e^{i θ} x)}^{n} w i t h n i n t e g r n a t u r a l$ $1) f i n d t h e r o o t s o f p (x)$ $2) f c t o r i z e i n s i d e C [x] p (x)$ $3) f a c t o r i z e i n s i d e R [x] p (x) . θ \in R$

Question Number 39053 Answers: 1 Comments: 0

Question Number 38933 Answers: 0 Comments: 9

[Refreshing an old question] Find the smallest number which is a multiple of 75 and has 75 divisors.

$[R e f r e s h i n g a n o l d q u e s t i o n]$ $F i n d t h e s m a l l e s t n u m b e r w h i c h i s$ $a m u l t i p l e o f 75 a n d h a s 75 d i v i s o r s .$

Question Number 38907 Answers: 1 Comments: 5

(√(c^2 +x^2 )) = 1+(c/x) how many complex roots ?

$\sqrt{c^{2} + x^{2}} = 1 + \frac{c}{x}$ $h o w m a n y c o m p l e x r o o t s ?$

Question Number 38825 Answers: 1 Comments: 4

Question Number 38786 Answers: 0 Comments: 5

Question Number 38696 Answers: 1 Comments: 0

Question Number 38636 Answers: 1 Comments: 0

Three variables u,v and w are related such that u varies directly as v and inversely as the square of w. If v increases by 15% and w decreased by 10%, find the percentage change in u.

$Three variables u, v and w are related such that$ $u varies directly as v and inversely as the square of$ $w . If v increases by 15 % and w decreased by 10 %,$ $find the percentage change in u .$

Question Number 38587 Answers: 2 Comments: 2

solve for x: e^x + e^x^2 + e^x^3 = 3 + x + x^2 + x^3

$solve for x : e^{x} + e^{x^{2}} + e^{x^{3}} = 3 + x + x^{2} + x^{3}$

Question Number 38570 Answers: 0 Comments: 2

Question Number 38477 Answers: 1 Comments: 0

Question Number 38459 Answers: 0 Comments: 7

x^2 (x−b^2 )+a^2 b^2 (x−a^2 )=0 Solve for x.

$x^{2} (x - b^{2}) + a^{2} b^{2} (x - a^{2}) = 0$ $S o l v e f o r x .$

Question Number 37991 Answers: 1 Comments: 4

1. Find the sum s_n =1+2x+3x^2 +4x^3 +...+nx^(n−1) Hence,or otherwise, find the sum Σ_(k=1) ^n k.2^k 2. Simplify the following i. Σ_(r=0) ^n (_(2r−1) ^(2n) ) ii.Σ_(r=0) ^n (−1)^r r(_r ^n ) iii.Σ_(r=0) ^n (−1)^r (1/(r+1))(_r ^n ) iv.Σ_(r=0) ^n (_(2r) ^(2n) ) v.Σ_(r=0) ^n (−1)^r (_(n−r) ^(n+1) ) 3.Find the sum Σ_(r=0) ^(n−k) (_k ^(n−r) ), where k=0,1,2,3,...,n

$1 . F i n d t h e s u m$ $s_{n} = 1 + 2 x + 3 x^{2} + 4 x^{3} + . . . + {n x}^{n - 1}$ $H e n c e, o r o t h e r w i s e, f i n d t h e s u m$ $\underset{k = 1}{\sum^{n}} k . 2^{k}$ $2 . S i m p l i f y t h e f o l l o w i n g$ $i . \underset{r = 0}{\sum^{n}} (_{2 r - 1}^{2 n})$ $i i . \underset{r = 0}{\sum^{n}} {(- 1)}^{r} r (_{r}^{n})$ $i i i . \underset{r = 0}{\sum^{n}} {(- 1)}^{r} \frac{1}{r + 1} (_{r}^{n})$ $i v . \underset{r = 0}{\sum^{n}} (_{2 r}^{2 n})$ $v . \underset{r = 0}{\sum^{n}} {(- 1)}^{r} (_{n - r}^{n + 1})$ $3 . F i n d t h e s u m$ $\underset{r = 0}{\sum^{n - k}} (_{k}^{n - r}), w h e r e k = 0, 1, 2, 3, . . ., n$