Let S = x^3 + ax^2 + bx + c : a,b,c N, a,b,c 20 be a set of polynomials.

Question

Let $S = \{x^3 + ax^2 + bx + c : a,b,c \in \mathbb{N},\ a,b,c \le 20\}$ be a set of polynomials. Then the number of polynomials in $S$ which are divisible by $x^2+2$ is

Accepted Answer

Since the cubic has leading coefficient 1, write
$x^3+ax^2+bx+c = (x^2+2)(x+n)$
Expanding the right side: $x^3 + nx^2 + 2x + 2n$.

Comparing coefficients: $n = a$, $b = 2$, $c = 2n = 2a$.

So $b=2$ is forced (1 choice, within $[1,20]$). For the pair $(a,c)$ with $c=2a$ and $c \le 20$, we need $a \le 10$, giving $a = 1,2,\dots,10$ --- 10 choices.

Total polynomials $= 1 	imes 10 = 10$.

Solutions · 1