Let zEC is root polynomial fE R[x]. Prove ī is is root polynomial f€R[x].

Q: Let f(x) = x³ + 6 E Z,[x]. Write f(x) as a product of irreducible polynomials over Z,.

A: Consider the given polynomial, Factor theorem for field: Let F be a field be an element, a…

Q: Let P(x) be the Lagrange interpolating polynomial of 1 f(x) = ; 2x² + 2 at the points xo = 0; x, = 1…

A: We will use La Grange interpolation

Q: Calculate the Taylor polynomials T3(x) and T3(x) centered at æ = T for f(x) = tan(x). T2(x) = T3(x)

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Q: In what way is the nth Taylor polynomial of f(x) at x = a likef(x) at x = a?

A: To write the way is the nth Taylor polynomial of f(x) at x = a

Q: Write the Taylor polynomial T5(x), for the function f(x)=cos(x)f(x)=cos(x) centered at x=0. T5(x)= ?…

A: Function given is- f(x)=cos(x) centered at x=0 We need to write the Taylor polynomial T5(x) for the…

Q: Find the third Tayler polynomial P, (x) f fxl= cos X at

A: Given function is  fx=cosxat x=π

Q: Let f(æ) = VE = ai. For æo = 2.6, #1 = 3.4, x2 = 3.9, 23 = 4, æ4 = 4.5, %3D • What is the maximum…

A: Given function is f(x)=x13 and points are x0=2.6,x1=3.4,x2=3.9,x3=,x4=4.5 So corresponding function…

Q: Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) %3D хе 4х, а%3D 0…

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Q: For f(r) sin(Tx) on [0, 1], find the Bernstein polynomials B1, B2, B3, B4. Simplify answers as much…

A: We will find out the required value.

Q: Let P(x) be the Lagrange interpolating polynomial of 3+3 at the points 0 I and a 3 If P(x) ed to spi…

A: We will first find the Lagrange polynomial

Q: Construct for the function the Lagrange inter polation polynomial y= sim Tx, taking Tthe' pointa Ho…

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Q: Let P(x) be the Lagrange interpolating polynomial of 1 f(x) = 3x² + 2 at the points x, = 0; x, = 1…

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Q: Let f(x) = e^x. Find the polynomial P2(x) of degree at most 2 such that P2(1) = f(1), P2(2) = f(2),…

A: We have to find the polynomial of degree 2 for the function fx=ex using the divided difference…

Q: Let f(x) = x, the second Taylor polynomial P2(X) about Xo = 2 is: Select one: O 1+ 3x + 6x? 08-12x +…

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Q: Construct the Lagrange interpolation Polynomial for the function Y= Sin Tx, the ponts =0, 2, =…

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Q: Calculate the Tayorl Polynomials T2(x )and T3(x) centered at x=a for f(x) = 14 sinx, a=pi/2

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Q: Use the second Taylor polynomial P2 (x) for f(x) = x² + In(1 + x) about ¤, = 0 and evaluate f…

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Q: Let f(x) = , defined on [0, +). Find the 10th Taylor (b) polynomial T1o(x) centered at a = 2. 3+x

A: Taylor series is nothing but expressing the infinitely differentiable function in terms of its…

Q: Write the third degree Taylor polynomial of f(x) about x = 3

A: Let f be a function with derivatives of all orders for all real numbers and…

Q: Let P(x) be the Lagrange interpolating polynomial of 1 f(x) = 2x2 + 3 at the points xo = 0; x1 = 1…

A: We will be using the Lagrange basis to interpolate f(x) using a quadratic polynomial.

Q: Consider the function f(x)=x+2x.f(x)=x+2x. Construct the Lagrange interpolation polynomial which…

A: Lagrange's interpolation formula for n points can be defined as:…

Q: Let P(x) be the Lagrange interpolating polynomial of 1 f(x) = at the points xo = 0; x, = 1 and x =…

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Q: Use the third Taylor polynomial P3(x) for f(x) = x ln? x about xo = 1 and evaluate f(2). 17 O 1 13…

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Q: Let p(x) be an irreducible polynomial in F[x], if p(x) divides r(x)s(x) for r(x), s(x) e F[x], then…

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Q: Use the second Taylor polynomial P₂(x) for f(x) = x² + ln(1 + x) about x = 0 and evaluate f f(x)dx.…

A: Given function is fx=x2+ln1+x. By using the second Taylor polynomial P2x for the given function fx…

Q: Consider a quadrature of the form 1 |æ|f(x)dx = ÷(f(-1) + 2f(0) + f(1)), (f(-1)+2f(0) + f(1)), show…

A: Solution: The objective is show that the given quadrature is exact for any polynomial f(x) of degree…

Q: Let the chor a cteristic polynomial A, Then det (A)=? O 1 O o any real number. Does not exist.

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Q: Find n such that |e − Tn(1)| < 10−8, where Tn is the nth Maclaurin polynomial for f (x) = ex

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Q: Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = 11 tan−1(x),…

A: Given,

Q: Find the degree three talor polynomial for f(x) =/x centered at a =4

A: Given function is, fx=x        centered at    a=4

Q: Consider f(x) e on [0, b] and its approximation by an inter- polating polynomial p2(x) at the evenly…

A: Given that f(x)=ex on [0,b]We have to approximate the above function in [0,b]i.e…

Q: Find the Taylor polynomial p, centered at a P2 8 for f(x) = Vx.

A: Taylors series of function f(x) at a

Q: If the polynomial P(r) is the nth Taylor's polynomial for f(r) about the point zo then the upper…

A: Given: If the polynomial pnx is the nth Taylor's polynomial for fx about the point x0 then the upper…

Q: Consider the function f(x) = x = [0,2]. Construct the , Lagrange interpolation polynomial which…

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Q: Let P(x) be the Lagrange interpolating polynomial of 1 f(x) == 2x² + 2 at the points x, = 0; x, = 1…

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Q: Prove using the generating function of the Legendre Polynomials: t2 [xP(x) – nP,(x)]t": (1- 2xt +…

A: We have to prove ∑n=0∞xPn'(x)-nPn(x)tn=t21-2xt+t232 We know the generating function of the Legendre…

Q: Find the taylor polynomials of orders 0,1,2 and 3 generated by f at x=a. f(x)=4/x, a=1 p0(x)=    ?

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Q: Find the Taylor polynomials p,, .., P4 centered at a = 0 for f(x) = 3 cos (3x) P;(x) = P2 (x) = P3…

A: The given function is fx=3cos3x at a=0. It is required to find the polynomial approximation when…

Q: Let f(x) = x* – 4x³ + 4x2 + 12x – 21. Given that 2 – V3i is a root of f(x), factor f(x) into a…

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Q: Consider the function f(x) = 12, x = [0, 2]. Construct the Lagrange interpolation polynomial which…

A: Given function is fx=x21+x2, x∈0,2. We have to construct the Lagrange's interpolation polynomial…

Q: Construct the Langvange inter po lation polynomial for the function y= Sin Tx, taking the points %3D…

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Q: P"n+1 (x) – xP", (x) = (n+1)P,(x) = P,(x) %3D -

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Q: Given the Taylor polynomial for f(x) at x = 3, find ƒ(³) (3) T₁(x) = 1 + x −3+2(x − 3)² + 2(x-3)³ -…

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Q: Q3. Prove that if g interpolates the function f at ro, T1, . .. , In-1 and if h interpolates f at…

A: We have to prove that if g interpolates the function f at x0,x1,…,xn-1 and if h interpolates f at…

Q: Let F(x) = e -514 dt. Find the MacLaurin polynomial of degree 5 for F(x). - 0.12 -5x* dx. e Sx4 Use…

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Q: The Maclaurin polynomial for the function f(x) = tan(x), n=3 is P3(x) = x + 3 True False

A: I am attaching image so that you understand each and every step.

Q: Consider the function f(x) = , TE [0, 2]. Construct the Lagrange interpolation polynomial which…

A: Numerical methods: Legranges interpolation of the given data points

Q: Let P(x) be the Lagrange nterpolating polynomial of 2x +3 3 IfP(x) is used to approximate f(x), then…

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Q: Let f(x) = x + x² + x + 1 E Z,[x]. Write f(x) as a product of irre- ducible polynomials over Z,.

A: Given: fx=x3+x2+x+1 The given polynomial can be factorized as a product two irreducible…