For the following monic polynomial P(z) = zn+an−1z"−¹+...+a₁z+ao, show that all the roots lie in the disk D(0, R), where R = /1+ |an−1 |² + . ... + |a1|2 + |ao|². n-1 (Hint: If P(z) = 0, then -z" = Σaz. You can use the Cauchy- Schwarz inequality to prove that |z| < R.)

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A: Irreducible polynomial and it's order

Q: Show that f(x)=x\power{2}-3 is the minimal polynomial of \sqrt{3} over Q,

A: Let            We  have to  find the  minimal  polynomial of  the    over Let Then,  Suppose  that

Q: 49) Let P(z) = ao + a1z +……. 1 order n at o∞. Consider + anz" be a polynomial of degree n. Show that…

A: Given that Pz=a0+a1z+a2z2+⋯+anzn be a polynomial of degree n.Then, an≠0. We have to show that 1P(z)…

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A:

Q: The coordinates of the P(x)=x³ – 2x2 + 3x polynomial with respect to the base B-[1, x, x?,x'] are as…

A:

Q: Let f(x) = x' + 2x + 4 and g(x) - 3x + 2 in Z,[x]. Then, upon dividing f(x) by glx), we get 7- The…

A: I am solving part7 &amp; part8 as you mentioned.

Q: Use the Residue Theorem to find the following (a) ez dz 22 – 1 J]z-1|=1 (b) e dz Izl=42 22 – 1

A: *concept:- Residue theorem.

Q: if the Legendre polynomials P.(x) = -1 then: Find 1) ,x* P,(2) dx 2)P'(-1) A) L P.) dx = and P,(1) -…

A:

Q: Find the root f(x) = x³ - 4x - 9 = 0 by bisection method €= 1 x 10-4,[2,3] in the interval : a-…

A:

Q: Determine if the following polynomials are irreducible or not k(1, y) = 1® – y° € Q!r, y]

A: Given:- k(x ,y)=x6-x7 ∈ Q x,y

Q: Below is a graph of function f(x). Which of the following could be the second Taylor polynomial of…

A: We have to find the Taylor second order polynomial.

Q: By using Ho (X) = 1 = 1 and HI (X)=2x in Hermit polynomials, calculate the value of Hg (1) ?

A: The Hermite polynomial is a particular set of orthogonal polynomials. According to the definition…

Q: 4) For Bernstein polynomial, B₁ (f; x) = Σk=o (k) x* (1 − x)"-kƒ (), we have B₁ (2t² + 1;x) = 2x² +…

A:

Q: Use the bisection method to find p3 for ƒ(x) = x² + x – 1 on [0, 2] .

A: Calculate the value of the function at the mid-point of the interval.

Q: Given a positive integer d, define Q[vā] = {s(Va) : f € Q-]}. i.e. the set of real numbers that can…

A:

Q: Let z1 = -2 + 11i, z2 = 2 – i. Showing the details of your work, find, in the form x + iy: Re (zi).…

A: Here we can use following formula If z1=a+bi, z2=c+di,…

Q: 1 1+ 12,2 over [-1, 1]. SET-UP the Lagrange (-1, 1). 2. Consider the function defined by f(x)…

A:

Q: 11. The Hermite polynomials, H,„(x), satisfy the following: į. = Se-**H„(x)Hm(x) dx = \T2"n! 8,m:…

A: To prove: ∫-∞∞xe-x2HnxHmxdx=π2n-1n![δm,n-1+2n+1δm,n+1] Proof: Take the LHS ∫-∞∞xe-x2HnxHmxdx and…

Q: 2)tind the third degree Taylor polynomial Is (x) centered at a= I forr the function f (x)= 2

A:

Q: In linear space, consider all true polynomial functions with internal multiplication x(t)y(t)dt for…

A: We will use GRAM-SCHMIDT ORTHOGONALIZATION PROCESS to solve this problem For a basis y0,y1,y2 to…

Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

A:

Q: f(x) = x² − 3 is the minimal polynomial of √3 over Q,

A: We have that              f(x) = x2-3  be a polynomial of two degree.

Q: a) Evaluate r(금). r(0)

A: According to our company policy, we are supposed to answer the only first question. For the rest…

Q: Find the splitting field for f (x) = (x2 + x + 2)(x2 + 2x + 2) overZ3[x]. Write f (x) as a product…

A: Given: fx=x2+x+2x2+2x+2. We have to find the splitting field of f(x) over Z3x.Also, write f(x) as a…

Q: z/(z-3)2 dz ; C: |z+3| =3 slove by residue theorem

A: The given integral is,