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)=−2x4+2x2+4
(1 point)
Responses
1 real solution, 3 imaginary solutions
1 real solution, 3 imaginary solutions
2 real solutions, 2 imaginary solutions
2 real solutions, 2 imaginary solutions
2 real solutions, 0 imaginary solutions
2 real solutions, 0 imaginary solutions
4 real solutions, 0 imaginary solutions
4 real solutions, 0 imaginary solutions
To determine the number and type of solutions for the polynomial function P(x) = -2x^4 + 2x^2 + 4, we can use the discriminant.
The discriminant is calculated using the equation b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, the polynomial P(x) is a quadratic function (-2x^4 + 2x^2) with no linear term (b = 0), and a constant term of 4 (c = 4). Therefore, the discriminant can be calculated as follows:
Discriminant = b^2 - 4ac
= 0^2 - 4(-2)(4)
= 0 + 32
= 32
Since the discriminant is positive (32 > 0), the quadratic equation has two real solutions.
Therefore, the correct answer is:
2 real solutions, 0 imaginary solutions.