provide a full handwritten solution to all parts

4. Polynomials as an Algebraic structure. This question is relevant to chapter 9. a) Let Mar) and r;{;rr) be two polynomials with real coefficients Prove that ifpl;r:)q(;ir) = l]. then either My) 2 [l b) d) or q(;ii) = l). (Hint: there is a very short argument by contradiction, which is based on the coefficients of the highest powers of 3:.) A polynOmial is said to be irreducible over R, if it cannot be factored to smaller degree polynomials of real coefficients. For example, all polynomials of degree 1 are irreducibles, and polynOmials of the form 1-2 + 0.2 are also irreducibles over R. Find a criterion to describe all the quadratic polynomials with real coefficients that are irreducible over R. The Fundamental Theorem of Algebra (of chapter 9) claims that any polynomial (with whatever type coefficients) has a root in C. Given a polynomial My) of degree in. > 1 with real coefficients. and assume 2., is a root of this polynomial. We have two possibilities: i) Case 1: an is real. Call it r and prove that p(:r:) = {:r: 'r')q{;r:) where q{;r:) is a polynomial of degree Ti. 1, also with real coefficients. ii) Case 2: 20 is not real (that is, 1111(4)) % 0). Prove that ME") = 0, and that 03(2) 2 (z zu)(z EU) is a quadratic polynomial (degree 2) with real coefficients. And finally prove 13(2) 2 d(z)q(z) where (1(a) is a polynomial of degree n 2 with real coefficients. (Hint: Use Theorem 9.3.6. Before you consider the cases, first show that if p(2:) and 05(2) are polynomials of real coefficients, and Me) = d(z)q(z) for some polynomial (1(2), then q(z) has real coefficients). Prove using PCMI that for any polynomial Mar) with real coefficients, p(.-r!) can be decomposed into products of linear and irreducible quadratic polynomials (both with real coefficients). Continue to conclude any polynomial with real coefficients can be decomposed into products of linear factors with coefficients in C 2 RH)