Let F be a field and a ∈ F \ {0} which is a zero of some polynomial f(x) ∈ F[x]. Find a polynomial g(x) ∈ F[x] with deg g = deg f whose leading coefficient is the constant term of f(x) and with zero 1/a.

Q: Consider the following vectors in R³: Give examples of vectors V1 Enter the vector w in the form…

A:

Q: Problem 7 Prove that the function f: Rk → R where f(x₁,... k) = 1; is continuous.

A:

Q: (#4) When is a function f(.): R" R convex and proper? Show that f(-) is convex and proper implies…

A: Function f : Rn→ R is convex if its domain is a convex set and for all x, y in its domain, and all λ…

Q: y = aebx

A:

Q: Solve for the determinant of the given matrices:

A: 1.4. Given, ∫x1=-1x2=26xdx=3x123x22.

Q: 10.2 Determine the chromatic numbers of the following graphs. Feel free to look up what the graphs…

A:

Q: For each sub-item, determine using a truth table if the given statement is true. a. ~(PAQ) and ~P…

A:

Q: We model pumping from spherical containers the way we do from other containers, with the axis of…

A: Given Data: Let us cut a slice of water of width ∆y, -y mfrom the top. so the radius of the slice…

Q: Find the measure of each angle of the triangle.   x °  (2x + 2)°  (3x - 2)°

A:

Q: Solve 371044 = ... (mod59)

A:

Q: 1. Consider the function f: RR given by f(x) = x¹ = 4x³+4x². (a) Using calculus, find all critical…

A:

Q: 1. Find all solutions: 88- 4. For which b are the vectors 2. In what ways can be written as a linear…

A:

Q: Minimize the objective function 10x+6y subject to the constraints. y   ≥ −2x+11   y ≤ −x+10 y ≤…

A: This problem is infeasible because these is no common point which satisfies all the constraints.

Q: Consider the following statements in English. Write a logical expression with the meaning. The…

A:

Q: You have to solve an equation f(x) = 0 with a precision below a defined value TOL. If |ƒ(x₁)| < TOL…

A: For a given equation f(x)=0 with a precision below defined value TOL then - You have found an…

Q: The graph of y = f(z) is shown in blue. Draw the graph of y = f(2), shown in red, by determining how…

A: We have to find the function of f(x/3) with the help of red and blue graph

Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

A:

Q: 8) The total expenses of a camping party were $72. If there had been 3 fewer persons in the party,…

A:

Q: x²-16 21. m 4- 23. lim 25. 91 x²-6x 5x-6 x² +64 x-8

A:

Q: A trough of length L has a cross section in the shape of a semicircle with radius r (See figure…

A: Given: The volume of water is, V=L0.5πr2-r2arcsinhr-hr2-h2, where L=10 ft, r=1 ft, and V=15 ft3. To…

Q: Let A = {a, b, c, d}and B = {r, s, t} Determine whether each of the following functions is…

A: Use the relevant definitions.

Q: Clyde is hiding under a dock and is planning to swim to the boat that is located 148 feet from the…

A: We will find distance and then divide it by velocity .  Note : 1st we need to convert velocity from…

Q: Find the vectors T, N, and B at the given point. r(t) = (8 cos(t), 8 sin(t), 8 In(cos(t))), (8,0,0)…

A: Introduction: In mathematics, a vector at a degree on a curve is uniquely decomposed because of the…

Q: Find the locus of the centers of the circles x² + y²-2ax-2by- +2=0, where a and b are parameters, if…

A:

Q: 1-7. Examine the graph below. Then, in a sentence or two, suggest reasons why the graph rises at…

A: There could be many reasons for the the price rise of purse. We will consider most affected reason…

Q: the line has a rate of change of -5 and y=3 when x=0

A: The rate of change of a function at a point is the change of the dependent variable y at the point…

Q: Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral…

A: Numerical integration methods can generally be described as combining evaluations of the integrand…

Q: Let D be an integral domain and suppose all ideals in D are principal. Prove that if p ∈ D is an…

A:

Q: Find the solution of each of the following differential equations. 3. y' = x+xy²; y(1) = 0 4y

A: Sol:- y' =x+xy24yA first order separable ODE has the form of Ny·y'=MxRewrite in the form of a first…

Q: Find the equations of the normals to the circle x² + y²-8x-2y + 12 = 0 at the points whose ordinate…

A:

Q: The interval from -2.07 to 1.87 captures an area of 0.95 under the z curve. This implies that…

A:

Q: 22. How many different committees of 6 can be formed from 12 boys and 8 girls if each committee is…

A:

Q: unity. Show, by example, that a maximal ideal in R need not be a maximal id

A:

Q: 8. (a) It is not known whether there are infinitely many prime pairs, i.e., odd primes whose…

A:

Q: Suppose y satisfies the initial value problem fy" (t) = 4y' (t) + 5y(t) = 8(t − 3) y(0) = 0 and…

A:

Q: Let A be a 2 x 3 matrix such that 4 √1 3 2 Find A 1 6 4 9 ^[] - [] [] - [8] A 2 = and A = 5 10 3

A:

Q: x8=24+2³+2+1 mod pras x² = N AES polynomial

A: As per the question we are given the following congruence equation for the AES function modulo P(x).…

Q: 3. Prove or disprove [yes, these should look familiar from §2.3]: (a) The sum of three consecutive…

A:

Q: Give all the associates of 2x − 7 in the ring (a) Z[x]; (b) Q[x]; and (c) Z_11[x].

A:

Q: In this problem, the domain is the set of all integers. Which statements are true? If an existential…

A:

Q: Directions: In each of the following problems, assume a, b, c and d are arbitrary constants,…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: The following sequences all converge to zero as n Find the rate of convergence of each sequence.…

A:

Q: 3. For the function f(z) = e, describe the image of the vertical line Rez= -1, and the image of the…

A: The given function is ,                  f(z) = ez and  image of vertical  line  Re z = -1  and…

Q: 8. dy (y+1)(x-2)(y-3) (x+1)(y-2)(x-3) = dx

A: given differential equation dydx=y+1x-2y-3x+1y-2x-3 using separation of variable…

Q: 1. Given x(t), x(t) = a. Draw x(t) b. Find and drawx(t) C. 1 -2 -1 -1≤t<1 1≤t<1.5. 2≤t<3 otherwise…

A: Given: xt=1, -1≤t&lt;1-2, 1≤t&lt;1.5-1, 2≤t&lt;30, otherwise   To find and draw: zt=∫-∞tyTdT, where…

Q: 2. Determine the inverse of the following matrix. Express your answer in rational form. Indicate the…

A:

Q: 9. det (CD+DC)

A:

Q: ÿ +6y=t²e-at + 4 = sin² t y(0) = 0 x(0) = 1 (0) = 0. (0) = 0.

A:

Q: Consider the matrix 2 2 1 3 3 3 8 # 2 5 5 9 0 3 4 A has λ = 1 as an eigenvalue. Calculate a…

A:

Q: Predicates P, T and E are defined below. The domain is the set of all positive integers. P(x) : x is…

A: Given,  P(x) : x is even . T(x,y) : 2x=y . E(x, y, z) : xy=z . To indicate each logical expression…