Polynomial Divisibility Counting Problem

Learn how to solve polynomial divisibility problems by using factor theorem, remainder theorem, and coefficient comparison techniques.

❓ Question

Let

S=\{x^3+ax^2+bx+c \; ; \; a,b,c\in\mathbb N,\ a,b,c\le 20\}

Then the number of polynomials in $S$ which are divisible by

x^2+2

is:

✍️ Short Explanation

If a cubic polynomial is divisible by $x^2+2$ , then:

x^3+ax^2+bx+c=(x^2+2)(x+k)

for some integer $k$ .

Now compare coefficients.

(x^2+2)(x+k)

= x^3+kx^2+2x+2k

Comparing with

x^3+ax^2+bx+c

gives

a=k,\qquad b=2,\qquad c=2k

Given:

a,b,c\in\mathbb N,\qquad a,b,c\le 20

Since

a=k

and

c=2k

we need

k\in\mathbb N

and

2k\le 20

k\le 10

Thus

k=1,2,3,\ldots,10

Total values of $k$ :

10

Each value gives a unique polynomial.

\boxed{10}

Option (2)

Whenever a cubic is divisible by a quadratic:

\text{Cubic}=(\text{Quadratic})(\text{Linear})

Write:

(x^2+2)(x+k)

and compare coefficients directly. This avoids using roots $\pm i\sqrt2$ and makes the counting question a 10-second problem.