Prove that ithe polynomial PCO =-2xP +2x2+loxx t6 has and fachor azero at M= 3, and fachm s ala the polynomi.

Q: (a) Consider the inner product on P2 defined by (p, q) = aobo + 2a1b1 + 3azb2 for each polynomial…

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Q: Find the linear polynomial p(x) = a1x + ao, that interpolates the two points with coordinates (3, 0)…

A: Given query is to find the equation of the line.

Q: A cubic spline S is defined by: So(x) = 3(x- 1) + 2(x – 1)² – (x – 1) S(x) = s,(x) = 4 + 4(x – 2) –…

A: Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously differentiable.…

Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: Suppose ƒ(0) = 1, ƒ'(0) = 2, and ƒ"(0) = -1. Find the quadraticapproximating polynomial for ƒ…

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Q: show that Q has infinitely many distinct quadratic extensions in C

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Q: The number of irreducible polynomials over Z13 of the form x + ajr + a2 is 91 O 182 O 78 O 169

A: Given x2+a1x+a2 is a 2nd degree polynomial .For a 2nd degree polymonial , no of irraducible…

Q: In a cubic spline approximation assume that we know that S;(x) = 4 – 5x+ 2x² + 3x³ Si+1(x) = a + bx…

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Q: fales, no4, s

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Q: Lat R= 7@7. Show that %3D the polynomial (1,0)x in has in finitely R[x] many roots in R

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Q: Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].

A: We have to show that 2 is equal to the product of a unit and the square of an irreducible in Zi. The…

Q: 2) FIND THE TAYLOR POLYNOMIAL 13 (x) foR THE FUNCTEON f CENTERED AT THE NUMBIER a . TOU ON -4x flx=…

A: f(x)=xe-4x, centre at a, where a=0

Q: Prove Viete's Formulas: The quadratic polynomial with ∝ and β as roots is     x2 +bx+ c, where b =…

A: Viete's formula relates the coefficient of the function with the roots of the function. Sum of the…

Q: Show that A = has two different factorizations of the form A = L · U where L is lower triangular…

A: Given: A=0001. To show: A has two different factorization of the form A=L·U, where L is lower…

Q: Apply Routh's Stability Criterion to the following polynomial to determine the condition for the…

A: For existence of stable roots 1. All the coefficient of polynomial must be positive 2. Number of…

Q: i) what ins the nE Taylor polynomial , la) of a fune bien fx) at x-c? What does Taylor's Theorem say…

A: We have to find the Taylor's polynomial pn(x) of a function at x=c

Q: Use a Taylor polynomial centered at x = 0 to estimate In(1.20) to within 0.01. a) 0.2 b) 00.18 c)…

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Q: Suppose X = (X1, X2, . .. , Xk) ~ Multinomial(n, p) , with p = (p1, P2, ..., Pk), prove that Cov(X;,…

A: Introduction :-  Given that X is a vector of random variable and it follows Multinomial(n,p) Where p…

Q: Find all the orginal funchians of the funckion fryl=2y (y-s)²

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Q: Find the Taylor polynomials p3 and p4 centered at a = 1 for ƒ(x) = 8√x.

A: Given f(x)=8x centered at a=1

Q: use lagrange multipliars to maxim The Fungtien 7CX>y)=V57-x" -y subjeet to the Following canstaint…

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Q: Find the Taylor polynomial p1 (x) for f(x) = 8x2 at x = 5. %3D Provide your answer below: P,(x) = O

A: To find the taylor polynomial p1(x) .

Q: Change the following data, such that the data follows a certain fourth-degree polynomial. X: -2 -1 0…

A: Newton's divided difference formula can be defined as:…

Q: A cubic spline S is defined by: So(x) = 3(x – 2) + (x – 2)² – 4(x – 2)ª 32 if 25x53 S(x) s,(+) =…

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Q: Verify that the Legendre polynomial Pn(x) satisfiesthe second-order equation(1 − x2)y'' − 2xy' + n(n…

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Q: Show that for any set of interpolant nodes, we have j=0 for all m € {0, 1, ., n}, where l,(r) is the…

A: Consider ,any set of k + 1 data points

Q: . Find the Taylor polynomial p3 centered at a = 4 for ƒ(x) = 2/x. %3D

A: To find the Taylor polynomial p3 centered at a=4 for the given function f(x)

Q: By using Ho(x)=1 and Hi (X)=2x in Hermit Polynomials, caleulate the Valne of Hg L)? (1)

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Q: Find the Taylor Polynomial of degree 3 and estimate the Remainder of in X0 = 0 for x = 0.1

A: Consider the given function whose taylor polynomial is required to calculate.

Q: Find the Taylor polynomial of orders 0,1,2,3 generated by f at x=a

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Q: Find the Taylor polinomials of orders 0, 1, 2, and 3 generated by f(x)= Vx+4 at a = 0.

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Q: 2) Find divergonce of the following functon Hoy,z) = xe ーZ ーD 472 リ+3yek

A: We will find out the required value.

Q: Find the integer a in Z5 so that (x-a) become a divider for f(x)=x^3+2x^2+3x+3 over Z5[x]

A: We have to find an a∈Z5 such that x-a is a divider for the polynomial fx=x3+2x2+3x+3 over Z5x. If…

Q: Find the nth Taylor polynomial centered at c. f(x) = In(x), n = 4, c = 6 P4(X) =>

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Q: Find the Radus of convegnce Rand the niterral of Convegua I of the Taylor senes (12, (x+1) n(2n=1)…

A: The given series is: ∑n=1∞ (x+1)2nn(2n-1) 4n

Q: Prove or disprove that there exist nonconstant polynomials in Zp[x] that are units if p is prime.

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Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

Q: Ques. Apply a change of vaviable X=Py txansforms Hhe quuardratic form 2. into a quadratic form with…

A: Consider  Rx=5x12+6x22+7x32+4x1x2-4x2x3 In matrix form: XTAX=x1x2x352026-20-27x1x2x3 Here,…

Q: 1. Use the Bisection method to find the point P3 for finding the roots of (x) = x – cosz on [0, 1].;

A: Find P3.

Q: Find the Taylor polynomials p3 and p4 centered at a = 0 for ƒ(x) = (1 + x)-3.

A: Given that: The function f (x) = ( 1+x) - 3 .

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

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Q: Sketch the bode plots for the complex conjugate Zeros in a quadratic factor for the under damped…

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Q: (a) Consider the inner product on P2 defined by (p, q) = aobo + 2a¡b1 + 3azb2 for each polynomial…

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Q: 5. Show that for Chebyshev polynomials (Tn, Tm) = 0, n #m, π, EIN n=m= 0, n = m = 0.

A: We have to solve given problems;

Q: Show that if P(z) is a polynomial of degree n then f(z) = is holomorphic on C \ P(z) {z1,..., zm},…

A: Given that Pz is a polynomial of degree n. Then by Fundamental theorem of algebra, Pz has n roots…

Q: ComPute the taylor polynomial ot order 2nd desree Poly nomial x=B, where A, B,w,Ø are Cun Stants for…

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Q: 7. Let z be a root of the polynomial p(z) = (z + 1)" + z". Show that %3D 1 Re z = - 2

A: For the solution follow the next step.

Q: Find the Taylor polynomials (centered at zero) of degrees 1, 2, 3, and 4. S f(x) = In(4x + 1)

A: Given data:     The given function is: f(x)=ln(4x+1). The given center is: a=0.     The expression…

Q: Use the error bound for SN to obtain another proof that Simpson’s Rule is exact for all cubic…

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Q: Let fa) = x -1E Zs CxJ, wrte fX) as a produt of ireduclble polynomias as Zg.

A: Answer