let-p-x-1-ix-n-1-ix-n-1-find-the-roots-of-p-x-and-factorize-p-x-give-p-x-at-form-of-arcs-

Question Number 30522 by abdo imad last updated on 22/Feb/18

l e t p (x) = {(1 + i x)}^{n} - {(1 - i x)}^{n}

1) f i n d t h e r o o t s o f p (x) a n d f a c t o r i z e p (x)

) g i v e p (x) a t f o r m o f a r c s .

Commented by abdo imad last updated on 24/Feb/18

1) w e h a v e p (x) = 2 i I m {(1 + i x)}^{n} b u t ∣ 1 + i x ∣= \sqrt{1 + x^{2}}

\Rightarrow 1 + i x = \sqrt{1 + x^{2}} (\frac{1}{\sqrt{1 + x^{2}}} + i \frac{x}{\sqrt{1 + x^{2}}}) = r e^{i θ} \Rightarrow r = \sqrt{1 + x^{2}}

a n d c o s θ = \frac{1}{\sqrt{1 + x^{2}}} a n d s i n θ = \frac{x}{\sqrt{1 + x^{2}}} \Rightarrow t a n θ = x

\Rightarrow θ = a r t a n x \Rightarrow {(1 + i x)}^{n} = (\sqrt{1 + x^{2})}^{n} e^{i n a r c t a n x} \Rightarrow

p (x) = 2 i {(\sqrt{1 + x^{2}})}^{n} s i n (n a r c t a n x)

p (x) = 0 \Leftrightarrow n a r c t a n x = k π \Leftrightarrow a r c t a n x = \frac{k π}{n} \Leftrightarrow x_{k} = t a n (\frac{k π}{n})

a n d k \in [[0, n - 1]] a n d p (x) = λ \prod_{k = 0}^{n - 1} (x - t a n (\frac{k π}{n})) l e t

f i n d λ w e h a v e p (x) = \sum_{k = 0}^{n} C_{n}^{k} {(i x)}^{k} - \sum_{k = 0}^{n} C_{n}^{k} {(- i x)}^{k}

= \sum_{k = 0}^{n} C_{n}^{k} ({(i)}^{k} - {(- i)}^{k}) x^{k} = 2 i \sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1} x^{2 p + 1}

\Rightarrow λ = 2 i C_{n}^{2 [\frac{n - 1}{2}] + 1} \Rightarrow

p (x) = 2 i C_{n}^{2 [\frac{n - 1}{2}] + 1} \prod_{k = 0}^{n - 1} (x - t a n (\frac{k π}{n})) .

2) p (x) = 2 i {(\sqrt{1 + x^{2}})}^{n} s i n (n a r c t a n x) .

Answered by sma3l2996 last updated on 23/Feb/18

p (x) = \sqrt{{(1 + x^{2})}^{n}} (e^{i n θ} - e^{- i n θ}) ∖ θ = {t a n}^{- 1} x

= \sqrt{{(1 + x^{2})}^{n}} (2 i s i n (n θ))

p (x) = 2 i \sqrt{{(1 + x^{2})}^{n}} s i n ({n t a n}^{- 1} (x))

{t a n}^{- 1} (x_{k}) = \frac{k π}{n} \Rightarrow x_{k} = t a n (\frac{k π}{n})

s o r o o t s o f p (x) a r e α_{k} = t a n (\frac{k π}{n}) ∖ k = (0, 1, 2, 3, \dots, n - 1)

p (x) = 2 i \sqrt{{(1 + x^{2})}^{n}} \prod_{k = 0}^{n - 1} (x - t a n (\frac{k π}{n}))

Commented by abdo imad last updated on 24/Feb/18

y o u h a v e c o m m i t e d a e r r o r i n l e a d i n g c o e f f i c i e n t o f p (x)

s i r .