Question

If the reminder is the same when polynomial p(x)=x^3+8x^2+17x+ax is divided by (x+2) and (x+1) then find the value of a

Answer 1

$\mathfrak{\huge{Answer:}}$

$\mathbb{GIVEN}$

Polynomial $\sf{p(x) = x^{3} + 8x^{2} + 17x + ax}$ leaves the same remainder when divided by $\sf{(x + 2)\:and\:(x + 1)}$.

$\mathbb{TO\:FIND}$

The value of (a) in p(x)

$\mathbb{METHOD}$

What we'll do is just divide p(x) by the factors one by one. Doing the same here, we get :

$\sf{p(x) = x^{3} + 8x^{2} + 17x + ax}$ divided by $\sf{x+1}$

x + 1 = 0

=》 x = (-1)

When we keep this value in p(x) :-

$\sf{(-1)^{3} + 8(-1)^{2} + 17(-1) + a(-1)}$

Solve this formed equation further

=》 $\sf{(-1) + 8 - 17 - a}$

=》 $\sf{(- a -10)}$

This will be equal to some remainder, which is unknown to us. We are even given that the remainder will be equal to the remainder left after dividing p(x) by (x+2). Divide them :-

$\sf{p(x) = x^{3} + 8x^{2} + 17x + ax}$ divided by $\sf{x+2}$

x + 2 = 0

=》 x = (-2)

When we keep this value in p(x) :-

$\sf{(-2)^{3} + 8(-2)^{2} + 17(-2) + a(-2)}$

Solve this formed equation further

=》 $\sf{(-8) + 32 - 34 - 2a}$

=》 $\sf{(-2a -10)}$

Keep these equal :-

$\sf{(-2a - 10) = (-a - 10)}$

Thus =》 $\tt{\huge{a = 0}}$

Answer 2

HeyMate❤....

i hope it will helps you...