Please solve question no 5 completely LTE7:29 PM 1 D ØVOLTE46%therTun-can be written as a product of linear factors z + a and quadraticfactors z? + az + b all of whose coefficients are real. (Use Problem24-7.)4. In this problem we will consider only polynomials with real coefficients.Such a polynomial is called a sum of squares if it can be written ashi? + ··· + hn² for polynomials h; with real coefficients.(a) Prove that if f is a sum of squares, then f(x) 0 for all x.(b) Prove that if f and g are sums of squares, then so is f g.(c) Suppose that f(x) 0 for all x. Show that f is a sum of squares. Hint:First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must beeven (why?), and g(x) 0 for all x. Now use Problem 3(b).5. (a) Let A be a set of complex numbers. A number z is called, as in the realcase, a limit point of the set A if for every (real) & 0, there is apoint a in A with |2 – al

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Please solve question no 5 completely LTE7:29 PM 1 D ØVOLTE46%therTun-can be written as a product of linear factors z + a and quadraticfactors z? + az + b all of whose coefficients are real. (Use Problem24-7.)4. In this problem we will consider only polynomials with real coefficients.Such a polynomial is called a sum of squares if it can be written ashi? + ··· + hn² for polynomials h; with real coefficients.(a) Prove that if f is a sum of squares, then f(x) > 0 for all x.(b) Prove that if f and g are sums of squares, then so is f g.(c) Suppose that f(x) > 0 for all x. Show that f is a sum of squares. Hint:First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must beeven (why?), and g(x) > 0 for all x. Now use Problem 3(b).5. (a) Let A be a set of complex numbers. A number z is called, as in the realcase, a limit point of the set A if for every (real) & > 0, there is apoint a in A with |2 – al < e but z # a. Prove the two-dimensionalversion of the Bolzano-Weierstrass Theorem: If A is an infinite subsetof {a, b] × [c, d], then A has a limit point in [a, b] × [c, d]. Hint:First divide [a, b] × [c, d] in half by a vertical line as in Figure 7(a).Since A is infinite, at least one half contains infinitely many points ofA. Divide this in half by a horizontal line, as in Figure 7(b). Continuein this way, alternately dividing by vertical and horizontal lines.(The two-dimensional bisection argument outlined in this hint is sostandard that the title &#34;Bolzano-Weierstrass&#34; often serves to describethe method of proof, in addition to the theorem itself. See, for example,H. Petard, “A Contribution to the Mathematical Theory of Big GameHunting,&#34; Amer. Math. Monthly, 45 (1938), 446-447.)(b) Prove that a continuous (complex-valued) function on [a, b] X [c, d]is bounded on [a, b] × [c, d]. (Imitate Problem 21-31.)(c) Prove that if f is a real-valued continuous function on [a, b] × [c, d],then f takes on a maximum and minimum value on [a, b] × [c, d].(You can use the same trick that works for Theorem 7-3.)eries*6.The proof of Theorem 2 cannot be considered to be completely ele-mentary because the possibility of choosing y with y&#34; = -a/ß dependson Theorem 24-2, and thus on the trigonometric functions. It is thereforeof some interest to provide an elementary proof that there is a solution forthe equation z&#34; – c = 0.(a) Make an explicit computation to show that solutions of 2? – c = 0can be found for any complex number c.(b) Explain why the solution of z&#34; – c = 0 can be reduced to the casewhere n is odd.(c) Let zo be the point where the function f(z) = z&#34; – chas its minimumabsolute value. If zo z 0, show that the integer m in the proof ofTheorem 2 is equal to 1; since we can certainly find y with y! =-a/B, the remainder of the proof works for f. It therefore suffices toshow that the minimum absolute value of f does not occur at 0.(d) Suppose instead that f has its minimum absolute value at 0. Since z is

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We will be solving Q.5 as mentioned.
Since you have asked multiple questions, we will solve the…