Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify as...

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

(a) Claim. Every polynomial of degree 3 with real coefficients has a real zero.

“Proof.” The polynomial p(x) = x³ – 8 has degree 3, real coefficients, and a real zero (x = 2). Thus the statement “Every polynomial of degree 3 with real coefficients does not have a real zero” is false, and hence its denial, “Every polynomial of degree 3 with real coefficients has a real zero,” is true.

(b) Claim. There is a unique polynomial whose first derivative is and which has a zero at

“Proof.” The antiderivative of 2x + 3 is x² + 3x + C. If we let p(x) = x² + 3x − 4, then p'(x) = 2x + 3and p(1) = 0. So p(x) is the desired polynomial.

c) Claim. Every prime number greater than 2 is odd.

“Proof.” The prime numbers greater than 2 are 3, 5, 7, 11, 13, 17, None of these are even, so all of them are odd.