If the remainder is the same when the polynomial \[p(x) = {x^3} + 8{x^2} + 17x + ax\] is divided by $(x + 2)$ and $(x + 1)$. Find the value of a.
A) 0
B) 1
C) -1
D) 3

Hint: We will first write down the Remainder Theorem for polynomials, which we will be using in the solution of this. Then, using that theorem, we will find the remainder of the polynomial on dividing by both the given polynomials and equate them.

Complete step-by-step answer:
Let us first write down the Remainder Theorem, which we will require in the solution.
Remainder Theorem: It says that if a polynomial p(x) is divided by (x - a), then we will get p(a) as the remainder.
Now, coming back to our question:
We have polynomials \[p(x) = {x^3} + 8{x^2} + 17x + ax\] which on division by $(x + 2)$ and $(x + 1)$ leave the same remainder.
For the first case, where we divide \[p(x) = {x^3} + 8{x^2} + 17x + ax\] by $(x + 2)$.
Comparing it to the definition of remainder theorem, we see that a = -2.
Hence, the remainder will be p(2).
Since, \[p(x) = {x^3} + 8{x^2} + 17x + ax\], so $p( - 2) = {\left( { - 2} \right)^3} + 8 \times {\left( { - 2} \right)^2} + 17 \times \left( { - 2} \right) + a \times \left( { - 2} \right)$.
On simplifying it, we will get:-
$ \Rightarrow p(2) = - 8 + 8 \times 4 - 34 - 2a$
On simplifying it further, we will get:-
$ \Rightarrow p(2) = - 8 + 32 - 34 - 2a = - 10 - 2a$ …………..(1)
For the first case, where we divide \[p(x) = {x^3} + 8{x^2} + 17x + ax\] by $(x + 1)$.
Comparing it to the definition of remainder theorem, we see that a = -1.
Hence, the remainder will be p(-1).
Since, \[p(x) = {x^3} + 8{x^2} + 17x + ax\], so $p( - 1) = {\left( { - 1} \right)^3} + 8 \times {\left( { - 1} \right)^2} + 17 \times \left( { - 1} \right) + a \times \left( { - 1} \right)$.
On simplifying it, we will get:-
$ \Rightarrow p( - 1) = - 1 + 8 \times 1 - 17 - a$
On simplifying it further, we will get:-
$ \Rightarrow p( - 1) = - 1 + 8 - 17 - a = - 10 - a$ …………..(2)
Since, we are given that the remainder is same when the polynomial \[p(x) = {x^3} + 8{x^2} + 17x + ax\] is divided by $(x + 2)$ and $(x + 1)$.
Hence, by comparing (1) and (2), which are the remainders of both, we will get:-
$ \Rightarrow - 10 - 2a = - 10 - a$
On rearranging the terms, we will get:-
$ \Rightarrow 2a - a = - 10 + 10$
On simplifying it, we will get:-
$ \Rightarrow a = 0$

Hence, the correct option is (A).

Note: The students must remember that they must know why the remainder theorem is working instead of trying to just learn this fact, they must know the validation behind it. Let us see the proof of Remainder Theorem:
Let us say we have a polynomial p(x) which needs to be divided by (x – a). Now, we need to show that the remainder is p(a).
Since, we know that $p(x) = (x - a)q(x) + r(x)$, where q(x) is the quotient and r(x) is the remainder.
We can write this as: $p(x) = (x - a)q(x) + r$.
Now, put x = a on both the sides in equation, we will get:-
$ \Rightarrow p(a) = (a - a)q(a) + r$
This is equivalent to:
$ \Rightarrow p(a) = 0 \times q(a) + r$
Hence, we have: $ \Rightarrow p(a) = r$.