Determine the number and type of solutions for the following polynomial function. Make sure to include an explanation in your work to justify how you determined your answer.
P (x) = - 2x^4 + 2x^2 + 4
(1 point)
The given polynomial function is a quartic function, which means it is of degree 4. To determine the number and type of solutions for this polynomial function, we can use the discriminant of a quadratic equation derived from the function.
Let y = 2x^2. This transforms the quartic function into a quadratic function:
-2y^2 + 2y + 4
The discriminant of this quadratic equation is:
Δ = b^2 - 4ac
Δ = 2^2 - 4*(-2)*4
Δ = 4 + 32
Δ = 36
Since the discriminant is positive (Δ > 0), the quadratic equation has two distinct real roots. This means that the original quartic function P(x) has four complex roots (counting multiplicities) according to the Fundamental Theorem of Algebra. So, the polynomial function P(x) = -2x^4 + 2x^2 + 4 has 4 complex roots.