Question

The number of polynomials of the form ${x^3} + a{x^2} + bx + c$ which are divisible by ${x^2} + 1$ and where a, b and c belongs to {1,2,…..10}, is
A.1
B.10
C.11
D.100

Answer 1

Hint: In this question, we need to determine the total number of possible polynomials in the form of ${x^3} + a{x^2} + bx + c$ such that it is completely divisible by another polynomial ${x^2} + 1$. For this, we will first apply the long division rule for the given polynomials and then, equate the remainder expression to zero so as to establish the relation between the constant terms.

Complete step-by-step answer:
$
  {x^2} + 1){x^3} + a{x^2} + bx + c(x + a \\
  {\text{ }}\underline {{}_ - {x^3} + {}_ - x{\text{ }}} \\
  {\text{ }}a{x^2} + (b - 1)x + c \\
  {\text{ }}\underline {{}_ - a{x^2} + {}_ - a{\text{ }}} \\
  {\text{ }}(b - 1)x + (c - a) \\
 $
According to the question, the polynomial ${x^3} + a{x^2} + bx + c$ should be divisible by the polynomial ${x^2} + 1$ , which implies that the remainder on division should be zero.
Here, on dividing the polynomial ${x^3} + a{x^2} + bx + c$ with the other polynomial ${x^2} + 1$ leaves a remainder of $(b - 1)x + (c - a)$ which should be necessarily zero.
Hence, the coefficient of x makes one relation, and the constant pair will make another relation such that,
$
  b - 1 = 0{\text{ and }}c - a = 0 \\
  b = 1 - - - - (i){\text{ and }}c = a - - - - (ii) \\
 $
The above two equations must be satisfied for the polynomial ${x^3} + a{x^2} + bx + c$ to be divisible by the polynomial ${x^2} + 1$.
Now, it is also given in the question that the possible values of the constants a, b and c are {1,2,3,4,5,6,7,8,9,10}. So, the total possible values of ‘a’ and ‘c’ are 10.
Hence, $c = a$ yields ten different possibilities for the polynomial ${x^3} + a{x^2} + bx + c$ to be drafted.
Therefore, the number of polynomials of the form ${x^3} + a{x^2} + bx + c$ which are divisible by ${x^2} + 1$ and where a, b and c belongs to {1,2,…..10}, is 10.
So, the correct answer is “Option B”.

Note: It is worth noting here that the value of the constant term ‘b’ always equals 1 whatever be the values of ‘a’ and ‘c’ and so it does not contribute to the total number of polynomials. Moreover, if the number of values of ‘b’ is more than one or it can also change then, the total number of polynomials will be calculated by adding all the possible values.