5. Using Rouch´e theorem to show that the polynomial z5 + 3z2 +
6z + 1 has exactly one
zero inside the circle @(1).
COMPLEX ANALYSIS
5. Using Rouch´e theorem to show that the polynomial z5 + 3z2 + 6z + 1 has exactly one zero insid...
Use the intermediate value theorem to show that the polynomial has a real zero between the given integers. f(x) = 4x3 - 2x - 5; between 1 and 3 Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. (Simplify your answers.) A. Because f(x) is a polynomial with f(1) = <0 and f(3) = <0, the function has a real zero between 1 and 3. B. Because f(x) is a polynomial with f(1)...
Show that the function flx)- x+8x+5 has exactly one zero in the interval [-1, 01. Which theorem can be used to determine whether a function f(x) has any zeros a given interval? O A. Extreme value theorem O B. Intermediate value theorem OC. Rolle's Theorem O D. Mean value theorem apply this theorem, evaluate the function fix)x +8x+5 teach endpoint of the interval [-1, 01 f-1)(Simplify your answer.) f(0) (Simplify your answer.) According to the intermediate value theorem, f(x) x...
Complex Analysis A and B plz A) B) = Use Rouche's Theorem to show that 24 + 4z +1 has exactly one zero inside |2| 1 Prove that all roots of z? – 523 + 12 = 0 lie between the circles [2] = 1 and |2| = 2
(i) Show that a non-zero polynomial in ??[?]Zp[x] has exactly ?−1p−1 associates. (ii) Let ?R be a field, 0≠?(?),?(?)∈?[?]0≠a(x),b(x)∈R[x]. Prove that ?(?)a(x) ?(?)b(x) are associates of each other if and only if ?(?)∣?(?)a(x)∣b(x)and ?(?)∣?(?)b(x)∣a(x). Q5 (4 points) (i) Show that a non-zero polynomial in Zp[x] has exactly p - 1 associates. a(x), b(x) E R[x]. Prove that a(x) b(x) are associates of each other if and only if a(x) | b(x) (ii) Let R be a field, 0 and b(x)...
Use the intermediate value theorem to show that the polynomial function has a zero in the given interval. f(x) = 2x® + 3x2 – 2x+8; (-8, -2] Find the value of f(-8). f(-8)= (Simplify your answer.) Find the value of f(-2). f(-2)= (Simplify your answer.) According to the intermediate value theorem, does f have a zero in the given interval? Yes Νο Ο
Using the complex-n-th roots theorem: 5. (a) Use Theorem 10.5.1: Complex n-th Roots Theorem (CNRT) to com- pute all the 4-th roots of -1/4. (b) Factor the polynomial 4x4 + 1 in C[x]. (c) Factor the polynomial 4x4 +1 in R[x]. (d) Use Rational Roots Theorem to prove that the polynomial 4x4 + 1 has no rational roots. Deduce the factorization of 4x4 + 1 in Q[x].
Multiplicity: Please use Write the equation of a polynomial that has exactly 2 distinct real zeros: one that is positive and one that is negative. Also, the positive multiplicity (but not 1)(use 3 or even multiplicity (but not 0) (use 2 or higher). You decide on the leading integer coefficient, only integers for all constants required! zero is to have odd higher) whereas the negative zero will have a 1. Write the polynomial equation below using the form:
Using the Intermediate Value Theorem explain why the function has at least one zero on the given interval, include as much detail as possible and work. Interval 87. Function h(x) = -2e-1/2 cos 22 T
Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z : zR with R appropriately chosen, (a) prove that pn(2) has (counting multiplicity) precisely n zeros in the open disc D(0, R); (b) also show that Pn(z) has no zeros in C \ D(0, R) Here you are asked to prove the Fundamental Theorem of Algebra...
Complex Analysis IVIatn 401: Homew ork Set # . 1. Apply the Cauchy-Goursat Theorem to show that Je f(z) dz 0 when the contour C is the unit circle with counterclockwise orientation, and when tl) /(z) t e2)tan . z- 3 222z 2