given the polynomial (x - 1)^2(1 + 2x^2), determine the following: a) the max number of real zeros. b) the…

Explanation:

Step1: Expand the polynomial

First, expand ((x - 1)^2(1 + 2x^2)). Since ((x - 1)^2=x^2-2x + 1), then ((x^2-2x + 1)(1 + 2x^2)=x^2\times1+x^2\times2x^2-2x\times1-2x\times2x^2+1\times1+1\times2x^2=2x^4-4x^3 + 3x^2-2x + 1). The degree of the polynomial (P(x)=2x^4-4x^3 + 3x^2-2x + 1) is (n = 4).

Step2: Determine the number of real - zeros

The maximum number of real zeros of a polynomial is equal to its degree. So, the maximum number of real zeros of the polynomial (P(x)) is (4).

Step3: Determine the number of x - intercepts

The (x) - intercepts of a polynomial function (y = P(x)) occur at the real zeros of the polynomial. So, the maximum number of (x) - intercepts is equal to the maximum number of real zeros, which is (4).

Step4: Determine the number of turning points

The maximum number of turning points of a polynomial function of degree (n) is (n - 1). For a polynomial of degree (n = 4), the maximum number of turning points is (4-1=3).

Answer:

a) 4 b) 4 c) 3