The-roots-of-the-fallowing-functions-are-the-Sequences-of-arithmetic-progressiyon-f-x-x-5-20x-4-ax-3-bx-2-cx-24-f-8-

Question Number 50445 by ANTARES VY last updated on 16/Dec/18

The roots of the fallowing

functions are the Sequences of

arithmetic progressiyon

f (x) = x^{5} - 20 x^{4} + {ax}^{3} + {bx}^{2} + cx + 24

f (8) = ?

Commented by tanmay.chaudhury50@gmail.com last updated on 16/Dec/18

p l s c h e c k t h e q u e s t i o n . .

i f f (x) = 0

v a l u e o f x i s i n t e r m s o f a, b, c

s o

f (8) = 8^{5} - 20 \times 8^{4} + a \times 8^{3} + b \times 8^{2} + c \times 8 + 24

Answered by ajfour last updated on 16/Dec/18

l e t r o o t s b e

x_{i} = p - 2 q, p - q, p, p + q, p + 2 q

\Rightarrow Σ x_{i} = 5 p = 20 \Rightarrow p = 4

Π x_{i} = p (p^{2} - q^{2}) (p^{2} - 4 q^{2}) = - 24

l e t q^{2} = s

\Rightarrow (16 - s) (16 - 4 s) + 6 = 0

4 s^{2} - 80 s + 262 = 0

o r 2 s^{2} - 40 s + 131 = 0

s = q^{2} = 10 \pm \frac{\sqrt{138}}{2}

(n o t a c o n v i n i e n t v a l u e . .)

a = Σ x_{1} x_{2}

b = - Σ x_{1} x_{2} x_{3}

c = Σ x_{1} x_{2} x_{3} x_{4}

f (8) = 8^{5} - 20 {(8)}^{4} + a {(8)}^{3} + b {(8)}^{2}

+ 8 c + 24

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