Question

Determine the roots of the following polynomial functions and describe each root. P(x)=(x^(2)+4)(x^(2)-1)

Answer 1

The roots of the polynomial function P(x) are x = 2i, x = -2i, x = 1, and x = -1, with the first two being imaginary and the last two being real.

Step-by-step explanation:

The provided polynomial function P(x) = (x2 + 4)(x2 - 1) can be factored to find its roots. This function is the product of two separate quadratic expressions, and we can find the roots of each expression separately.

For the first expression, x2 + 4, there are no real roots because the sum of a square and a positive number cannot be zero. Therefore, the roots are complex or imaginary. Specifically, they are x = 2i and x = -2i, where i represents the imaginary unit.

For the second expression, x2 - 1, this factors further into (x + 1)(x - 1), which means its roots are x = 1 and x = -1).

Therefore, the roots of the polynomial function P(x) are x = 2i, x = -2i, x = 1, and x = -1, with the latter two being real roots and the former two being complex roots.