Verified answer

Step-by-step explanation:

To find the zero of the polynomial F(x) = ak²x² + bkx + c, we can use the fact that if a and β are the zeros of P(x), then P(a) = P(β) = 0.

So, we have:

P(a) = a²a + ba + c = a³ + ba + c = 0

P(β) = a²β + bβ + c = 0

Multiplying the first equation by k² and substituting β for a in the second equation, we get:

k²a³ + k²ba + k²c = 0

k²β³ + k²bβ + k²c = 0

Subtracting these two equations, we get:

k²(a³ - β³) + k²b(a - β) = 0

Now, we can use the identity a³ - β³ = (a - β)(a² + ab + β²) to simplify the equation:

k²(a - β)(a² + ab + β²) + k²b(a - β) = 0

k²(a - β)(a² + ab + β² + b) = 0

Since a and β are the zeros of P(x), we know that a² + ba + c = β² + bβ + c = 0. Substituting these values, we get:

k²(a - β)(-b) = 0

Therefore, the zero of the polynomial F(x) is either a or β, depending on which one is different from -b/k².