MATH 5610/6860HOMEWORK #3, DUE THU SEP 24Notes: There are no extra-credit problems this time.1. B&F 2.4.2 and 2.4.4 (comparison of Newton’s method with and with-out modification for multiple roots)2. B&F 2.2.10 and 2.5.10 (Approximation of3√25 using fixed point iter-ation and Steffensen’s method. Please do not compare with bisectionmethod as is asked in 2.2.10).3. B&F 2.6.2 a,h. In this problem you are asked to find approximationsto all roots (real or complex) of a polynomial using the specializedversion of Newton’s method we saw in class (which uses Horner’smethod to evaluate p(z0) and p0(z0) efficiently). You can proceed asfollows:(a) Use different initial guesses to find all the real roots. Pleaseclearly indicate your initial guesses and the root it leads to.Example: In 2.6.1 d starting at z0= 1 we get approximate root1.12412. Starting at z0= 0 we get root −0.87605.(b) Then deflate the polynomial to remove all of it’s real roots usingHorner’s algorithm. In all the examples you have there will beat most one pair of complex conjugate roots that you can findby the usual quadratic formula. For example for the polynomialp(z) in 2.6.1 d deflating (i.e. dividing by the factors z −1.12412and z + 0.87605 and ignoring the small residual terms) we getthat p(z) ≈ (z − 1.12412)(z + 0.87605)q(z) where q(z) = z2+0.24807z + 3.04632 which has roots ≈ −0.12403 ± i1.74096.(c) If you want to double check your roots you can use the Mat-lab command roots. For example the roots of the polynomialp(z) = z4+ 2x2− x − 3 in B&F 2.6.1 d are given by:>> roots([1 0 2 -1 -3])ans =-0.1240 + 1.7410i-0.1240 - 1.7410i1.1241-0.876112 MATH 5610/6860 HOMEWORK #3, DUE THU SEP 244. K&C 3.5.1–3.5.3. Let p(z) = 3z5− 7z4− 5z3+ z2− 8z + 2. Pleasedo the following by hand.(a) Use Horner’s algorithm to find p(4).(b) Find the Taylor expansion of p(z) about the point z0= 4 (seeclass notes, this can be done by applying Horner’s algorithmsuccessively)(c) Start Newton’s method at the point z0= 4. What is the nextiterate z1?5. K&C 3.5.11, 3.5.12 Recall that any polynomial of degree n can bewritten asp(z) = c(z − z1)(z − z2) . . . (z − zn)and that the multiplicity of a root is the number of times it is re-peated in the factored form of p(z). Please show the following state-ments without using Thm 2.10 or Thm 2.11 in the textbook.(a) If z∗is a root of multiplicity m of p(z) thenp(z∗) = p0(z∗) = . . . = p(m−1)(z∗) = 0but p(m)(z∗) 6= 0. Hint: You may use Leibniz’s generalizedproduct rule:(fg)(n)(z) =nXk=0n!k!(n − k)!f(k)(z)g(n−k)(z).(b) The converse of previous statement. Hint: Use Taylor expan-sion of p(z)about