Let be a set of polynomials. Then the number of polynomials in

Question

Let $S = {x^{3} + a x^{2} + b x + c : a, b, c \in ℕ and a, b, c \leq 20}$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $x^{2} + 2$ , is [2026]

Smart Achievers · Accepted Answer

(1)

$x^{3} + a x^{2} + b x + c = (x^{2} + 2) (x + \frac{c}{2})$

$x^{2} : a = \frac{c}{2}$

$x : b = 2$

$b = 2, a = \frac{c}{2}, c \in {2, 4, \dots, 20}$

$Number of polynomials in S will be 10 .$

Solution

Q.
Let $S = {x^{3} + a x^{2} + b x + c : a, b, c \in ℕ and a, b, c \leq 20}$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $x^{2} + 2$ , is [2026]

1 10

2 20

3 120

4 6

Ans.
(1)

$x^{3} + a x^{2} + b x + c = (x^{2} + 2) (x + \frac{c}{2})$

$x^{2} : a = \frac{c}{2}$

$x : b = 2$

$b = 2, a = \frac{c}{2}, c \in {2, 4, \dots, 20}$

$Number of polynomials in S will be 10 .$

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