Question

If, x^4 + 4x^3 + 6ax^2 + 6bx + c is divisible by x^3 + 3x^2 + 9x + 3. Then, what is the value of a + b + c?

(a) 4

(b) 6

(c) 7

(d) 10

The question was asked in a job interview.

I want to ask this question from Applications of Quadratic Equations in portion Complex Numbers and Quadratic Equations of Mathematics – Class 11

Answer 1

The correct answer is (c) 7

To explain I would say: Here, f(x) = x^4 + 4x^3 + 6ax^2 + 6bx + c so, let its roots be, α, β, γ, δ and

g(x) = x^3 + 3x^2 + 9x + 3 so, let its roots be, α, β, γ

So, from here we can conclude,

α + β + γ + δ = -4 and α + β + γ = -3

Thus, δ = -1

This means (x + 1)(x^3 + 3x^2 + 9x + 3)

On solving this equation in simpler form we get,

x^4 + 4x^3 + 12x^2 + 12x + 3

=> 6a = 12 => a = 2

=> 6b = 12 => b = 2

=> c = 3

=> a + b + c = 7