The fundamental theorem of algebra asserts that any polynomial p(z) with complex coefficients may be written...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
The fundamental theorem of algebra asserts that any polynomial p(z) with complex coefficients may be written in the form p(z) = a(z-21) (z - 22). (z-Zn), where a is a nonzero complex number, n is the degree of p, and z, are the (not necessarily distinct) roots of p(z). Let us assume that n ≥ 2, and all the z; are distinct. We are going to show n i=1 (you do not need to prove this). 4. Prove that For illustration, consider the case n = 2 first, so we may write p(z) = az² + bz+c. The condition that the roots should be distinct is equivalent to the dicriminant 6² - 4ac being nonzero. Hence conclude that 1 p'(zi) 1. Write down the roots 21, 22 of p(z) (in terms of a, b, c). Verify directly that (₁) + (2₂) = 0. Now we return to the general case, where the degree of p is n ≥ 2. 2. Show that the derivative of p evaluated at 2, is given by p' (zj) = a(zj — 2₁) (zj — 2₂) (zj — Zj-1) (zj — Zj+1) ··· (zj — Zn) (Here we are assuming that p(z) = a(z-2₁) (2-22) (z-Zn) as stated at the top of the page.) The quantity is a times the product of (z; - zi) for all i different from j; notice that the term (zj zj) has been omitted. 3. Let , be a closed curve enclosing z, (where j is an integer from 1 to n) and no other z₁. Prove that Jag = 0. p(z)` In the last few problem sets we have dissected various regions of integration so that they contain only one singularity. The outcome has been that the integral over a large region is equal to the sum of smaller integrals, each of which may be computed (e.g. by the Cauchy Integral Formula). We are still working our way towards a theorem that does not require this dissection. In any case, we will let the curve I be the circle centred at 0 with radius R large (larger than any zi]). Then it turns out that we can similarly dissect I into closed curves, each of which contains one z; in the interior, the integral of which was computed in the previous part. So we get -dz = n 1 -dz. √p(2) d= = Σ/₁₁ (2) d dz 2=1 lim R→∞ Jr P(2) 2πi p'(zj) p(z) j 1 1 p'(zi) dz = 0. = 0. Suppose that p(z) has degree 1, so that there is one root, 2₁. Then in fact, p) #0 (i.e the result does not hold). So the assumption that n ≥ 2 really was necessary. The fundamental theorem of algebra asserts that any polynomial p(z) with complex coefficients may be written in the form p(z) = a(z-21) (z - 22). (z-Zn), where a is a nonzero complex number, n is the degree of p, and z, are the (not necessarily distinct) roots of p(z). Let us assume that n ≥ 2, and all the z; are distinct. We are going to show n i=1 (you do not need to prove this). 4. Prove that For illustration, consider the case n = 2 first, so we may write p(z) = az² + bz+c. The condition that the roots should be distinct is equivalent to the dicriminant 6² - 4ac being nonzero. Hence conclude that 1 p'(zi) 1. Write down the roots 21, 22 of p(z) (in terms of a, b, c). Verify directly that (₁) + (2₂) = 0. Now we return to the general case, where the degree of p is n ≥ 2. 2. Show that the derivative of p evaluated at 2, is given by p' (zj) = a(zj — 2₁) (zj — 2₂) (zj — Zj-1) (zj — Zj+1) ··· (zj — Zn) (Here we are assuming that p(z) = a(z-2₁) (2-22) (z-Zn) as stated at the top of the page.) The quantity is a times the product of (z; - zi) for all i different from j; notice that the term (zj zj) has been omitted. 3. Let , be a closed curve enclosing z, (where j is an integer from 1 to n) and no other z₁. Prove that Jag = 0. p(z)` In the last few problem sets we have dissected various regions of integration so that they contain only one singularity. The outcome has been that the integral over a large region is equal to the sum of smaller integrals, each of which may be computed (e.g. by the Cauchy Integral Formula). We are still working our way towards a theorem that does not require this dissection. In any case, we will let the curve I be the circle centred at 0 with radius R large (larger than any zi]). Then it turns out that we can similarly dissect I into closed curves, each of which contains one z; in the interior, the integral of which was computed in the previous part. So we get -dz = n 1 -dz. √p(2) d= = Σ/₁₁ (2) d dz 2=1 lim R→∞ Jr P(2) 2πi p'(zj) p(z) j 1 1 p'(zi) dz = 0. = 0. Suppose that p(z) has degree 1, so that there is one root, 2₁. Then in fact, p) #0 (i.e the result does not hold). So the assumption that n ≥ 2 really was necessary.
Expert Answer:
Answer rating: 100% (QA)
1 we write down the roots 2 2 plz and verify directly that them i riz a plz ... View the full answer
Related Book For
Fundamentals of Momentum, Heat and Mass Transfer
ISBN: 978-1118947463
6th edition
Authors: James Welty, Gregory L. Rorrer, David G. Foster
Posted Date:
Students also viewed these mathematics questions
-
Show that the (25-11) may be written in the form + (V-Pav) DABV 3D r
-
The gamma distribution may be written in several different (but mathematically equivalent) forms. Excel uses the following form for the two-parameter gamma distribution in its functions GAMMADIST...
-
The full model of Example 2.3 may be written in logarithmic terms as lnG/pop = + p ln Pg + y lnY + nc ln Pnc + uc ln Puc + pt ln Ppt + year + d ln Pd + n ln Pn + s ln Ps + . Consider the hypothesis...
-
Show that plane stress displacements for the Flamant problem in Section 8.4.7 under only tangential force X are given by: Data from section 8.4.7 My (1 v) - - -0 sin (1 v)X, -0 cost + 2X, 20 log r...
-
Eloise contributes $40,000 to MeldCo in exchange for a 30% ownership interest. During the first year of operations, MeldCo earns a profit of $200,000. At the end of that year, MeldCo holds...
-
On January 1, 2014, the Kobe Construction Company entered into a three-year contract to build a dam. The original contract price was $21,000,000 and the estimated cost was $19,400,000. The following...
-
Discuss why it is important to report significant changes in a patients condition to the treating physician.
-
Identify five recommendations made to strengthen the independent audit function following the Enron scandal. For each of these recommendations, indicate why you support or do not support the given...
-
Assume that ABC Insurance Company has purchased from QS Reinsurance Company a quota share treaty with a $500,000 limit and a retention of 30 percent and a cession of 70 percent. ABC has written...
-
You are to develop a simple Binary Search Tree ADT and run it against a test program. Avoid the temptation of finding code online. I am aware of all the available solutions and will be looking...
-
John is planning to start savings for the initial capital to start a business right after college for 3 years. John is expecting to get a job with a base salary of $85,000 payable with equal payments...
-
How does the term regulation relate to McDonalds and the other restaurants in the fast food industry? What kind of regulations do they have and who is the governing authority?
-
How do the size, technology, and mission of your organization (or college) directly affect you? Who are your organization's (or college's) competitors? (b) What changes do you see in information...
-
How would you solve this stimulation and stay within The budget of $42,500? like how many teams, skill level, outsourcing? Meeting- one on one coaching - how many hrs weekly, also with stand-up...
-
Describe how a successful compensation system should be developed for a company
-
The money paid by private business to the suppliers of loans used to purchase capital or money that households receive on savings accounts is called Blank______. Multiple choice question. interest...
-
Provide an example working as a case manager to , A company that manages diversity effectively promotes and confronts differences from the application process into leadership roles by inclusive...
-
As you rewrite these sentences, replace the cliches and buzzwords with plain language (if you don't recognize any of these terms, you can find definitions online): a. Being a jack-of-all-trades, Dave...
-
A cryogenic fluid flows in a 20-mm-diameter tube with an outer surface temperature of 75 K and an emissivity of 0.2. A larger tube, having a diameter of 50 mm, is concentric with the smaller one....
-
The maximum blood pressure in the upper arm of a healthy person is about 120 mm Hg (this is a gauge pressure). If a vertical tube open to the atmosphere is connected to the vein in the arm of a...
-
Ozone (O 3 ) dissolved in water is used in many wastewater treatment applications. Pure, 100% ozone gas at 1.0 atm and 20C is continuously bubbled into a tank of liquid water. The rising bubbles...
-
According to research on ethics in the workplace, _________ is/are often a major and frequent source of pressures that create ethical dilemmas for people in their jobs. (a) declining morals in...
-
A business owner makes a decision to reduce a plants workforce by 10% in order to cut costs and be able to save jobs for the other 90% of employees. This decision could be justified as ethical using...
-
If a manager fails to enforce a late-to-work policy for all workersthat is, by allowing some favored employees to arrive late without penaltiesthis would be considered a violation of _________. (a)...
Study smarter with the SolutionInn App