Question

p(x)=ax^3+ 3x ^2 - 3 and q(x)=2x^3-5x + a when divided by (x-4) leaves remainder r1 and r2 . find a if r1+r2= 0 .factorise the polynomial

Answer 1

$\textbf{\huge{ANSWER:}}$

$\sf{Given:}$

p ( x ) = $ax^{3} + 3x^{2} - 3$

q ( x ) = $2x^{3} - 5x + a$

p ( x ) and q ( x ) when divided by ( x - 4 ), give $r_{1}$ and $r_{2}$ as remainders respectively

$r_{1} + r_{2} = 0$

$\tt{Case\:1 : When\:p(x)\:is\:divided\:by\:(x-4) }$

=》 $a(4)^{3} + 3(4)^{2} - 3 = r_{1}$

=》 $64a + 48 - 3 = r_{1}$

=》 $\boxed{64a + 45 = r_{1}}$ ...(1)

$\tt{Case\:2 : When\:q(x)\:is\:divided\:by\:(x-4) }$

=》 $2(4)^{3} - 5(4) + a = r_{2}$

=》 $128 - 20 + a = r_{2}$

=》 $\boxed{108 + a = r_{2}}$ ....(2)

(1) + (2) = 0

=》 64a + 45 + 108 + a = 0

=》 65a + 153 = 0

=》 65a = (-153)

=》 $\boxed{a = \frac{(-153)}{65}}$

Suggestions : In these types of questions, you first need to note down all the information given in the question, as we did in the very starting of the answer. Then, you need to see how how can co-relate the information and then finally solve it. You can go through the answer again for more understanding and do ask the doubts in the comments section. :)