Given-a-polynomial-f-x-where-f-x-x-3-ax-2-3x-b-and-x-1-is-a-factor-of-f-x-and-f-x-12-at-point-2-4-find-the-values-of-a-and-b-hence-factorise-f-x-completely-

Question Number 38910 by Rio Mike last updated on 01/Jul/18

G i v e n a p o l y n o m i a l f (x) w h e r e

f (x) = x^{3} + a x^{2} + 3 x + b

a n d (x - 1) i s a f a c t o r o f f (x)

a n d f^{'} (x) = 12 a t p o i n t (2, 4)

f i n d t h e v a l u e s o f a a n d b

h e n c e f a c t o r i s e f (x) c o m p l e t e l y

Answered by ajfour last updated on 01/Jul/18

1 + a + 3 + b = 0

f^{'} (x) = 3 x^{2} + 2 a x + 3

f^{'} (2) = 12 + 4 a + 3 = 12

\Rightarrow a = - \frac{3}{4}; b = - \frac{13}{4}

f (x) = (x - 1) (x^{2} + p x + q)

\Rightarrow - q = b = - \frac{13}{4}

c o e f f i c i e n t o f x^{2}

p - 1 = a = - \frac{3}{4}

f (x) = (x - 1) (x^{2} + \frac{x}{4} + \frac{13}{4}) .

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