Quadratic Equations
Polynomial Divisibility — Counting Cubics
nta_pyq_2026_jan
Grade 11
Question:
Let $S=\{x^3+ax^2+bx+c:a,b,c\in\mathbb{N}$ and $a,b,c\leq20\}$ be a set of polynomials. Then the number of polynomials in $S$, which are divisible by $x^2+2$, is
Step-by-Step Solution
Key Concept: If $x^2+2$ divides $x^3+ax^2+bx+c$, write $x^3+ax^2+bx+c=(x^2+2)(x+k)$. Expanding: $b=2$, $c=2k=2a$.
10 polynomials.
Correct Answer: 1