Given that the expression 2x^3 +px^2 −8x+9 is exactly divisable by x^2 −6x+5, find the value of p and q. Hence factorise the expression fully


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Question Number 85490 by oustmuchiya@gmail.com last updated on 22/Mar/20

$G i v e n t h a t t h e e x p r e s s i o n 2 x^{3} + {p x}^{2} - 8 x + 9 i s e x a c t l y d i v i s a b l e b y x^{2} - 6 x + 5, f i n d t h e v a l u e o f p a n d q . H e n c e f a c t o r i s e t h e e x p r e s s i o n f u l l y$
Commented by john santu last updated on 22/Mar/20

$n o t h i n g q i n t h i s e q u a t i o n .$ $q u e s t i o n w r o n g!$
Commented by john santu last updated on 22/Mar/20

$i f d o y o u m e a n$ $2 x^{3} + {p x}^{2} - 8 x + q$ $b y r e m a i n d e r t h e o r e m$ $2 x (6 x - 5) + p (6 x - 5) - 8 x + q$ $12 x^{2} - 10 x + 6 p x - 5 p - 8 x + q$ $12 (6 x - 5) + 6 p x - 18 x - 5 p + q$ $72 x - 60 + 6 p x - 18 x - 5 p + q$ $(54 + 6 p) x + q - 5 p - 60 \equiv 0 x + 0$ $\Rightarrow 6 p = - 54 \Rightarrow p = - 9$ $\Rightarrow q = 60 + 5 p = 60 - 45 = 15$
Commented by john santu last updated on 22/Mar/20

$t h e n f a c t o r i s e$ $2 x^{3} - 9 x^{2} - 8 x + 15 = (x^{2} - 6 x + 5) (2 x + 3)$
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