10-If p is a prime number, then in Z[x] the polynomial x -1 –1 = a) (x - 1) (x + 2) (x –- 3)... (x + p – 1). b) (x – 1) (x² – 1) (x²– 1)... (L-- – 1). c) (x – 1) (x – 2) (x – 3)... (x – p + 1). d) (x 1) (x- 2) (x-3)... (x 1)

Q: 1. Consider y-f(x)=cos(2x)+x² over [0, 3t/2]. (By hand but use a calculator) a) Use the three nodes…

A:

Q: 2. Find a Taylor polynomial of degree N=5 for f(x) xpanded about xo=0.

A: Formula

Q: 8. Find a polynomial of degree 4 with integer coefficients and zeroes 3i and 4, with 4 a double…

A:

Q: 39.. Find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. 1 f(x) =…

A: We know that Taylor Expansion : f(x)=11+x=1-x+x2-x3+x4-.........∞.

Q: (e) +x-1 in Ca] 2. Prove that every nonzem f(x) e Flx] has a unique monie asnciate in F1x. 3 List…

A: Given that both are field hence we can Find corresponding unit of field then easily we can find its…

Q: 3. Find the fourth degree Taylor polynomial centered at C=8 for the function. f(x) =ln x In 8-x-8)…

A:

Q: -1 -2 Find the characteristic polynomial of A %3D 4 3 O 2 – 51 – 2 O 22 – 4X – 5 OX2 – 21 +5 OX2 +…

A:

Q: Determine the third and fourth Taylor polynomials of x^3 + 3x - 1 at x = -1.

A: To determine the third and fourth Taylor polynomials of  x3+3x-1 at x=-1

Q: 'hich of the following is the 3rd degree polynomial expansion of e" at a = 3? A. 0.453x³ + 8.34x –…

A: Let's find.

Q: 15. Find a polynomial P(x) of degree < 3 for which P(1) = y2 P'(1) = y2 P(0) = y1, P'(0) = y, with…

A: cubic polynomial is of form y=ax³+bx²+cx+d

Q: 4. Consider the polynomial f(x) = 4.x – 3x + 2. Note that f(-1) 1, and f(1) = 3. Determine if there…

A:

Q: The number of polynomials of degree 3 in Z5[x] is: O (4^4)*5 O 5^4 O 4^4 O 4*(5^3)

A:

Q: 1. Using (2n +1)Pn= Pn+1-P-1: prove that P(x) = (2n-1) Pn-1(x) + (2n-5) Pn-2+(2n-9) Pn-3+.... - and…

A: Given : 2n+1Pn=Pn+1'-Pn-1' To Prove : (a) Pn'x=2n-1Pn-1x+2n-5Pn-2+2n-9Pn-3+ . . . . . .…

Q: 9. Find a fourth-degree polynomial with integer coefficients that has zeros 3i and –1, with -1 a…

A: 3i and -1, with -1 a zero of multiplicity 2

Q: 9. Suppose that P(z) = ao + a,z + …+ anz" be a polynomial of degree n, where n is a positive…

A:

Q: 2.33 Let p(r) = x4 + x³ -- 2x2 + x + 1. Prove that the polynomial equation p(x) = 0 has a root in…

A:

Q: 1 Expand in ascending powers of x, up to and including the term in x', simplifying the V(1 –2x)…

A: To solve the following problem

Q: 13 - What is Taylor polynomial with order 3 generated by f(x) = x° + 3x² + 2x + 4 at x = 1.

A: The Taylor polynomial of order 3 is given as follows :

Q: ] 5. Consider h(x) = (x³ + 2)². (a) Expand h and compute h' using the power rule and polynomial…

A: Explanation of the answer is as follows

Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

A:

Q: .Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a…

A: Monic Polynomial: "Mono" stands for "Single" Therefore it is a type of polynomial having single…

Q: consider the function y=ax2+bx+c, where a is in the set {n∈ℕ|(n|15)}, b is an integer less than 10,…

A:

Q: 4. Suppose n is a positive odd integer. Prove that f(x) in Q[x]. (Hint. Assume the contrary and…

A: The given polynomial is f(x)=(x-1)(x-2)...(x-n)-1. Degree of given polynomial is n. Let us prove…

Q: 1. Using x, = 1,x1 = 2,x2 = 3, and x3 = 4, find the Lagrange interpolating polynomial of In(x + 1).

A: Given function is f(x)=ln(x+1) Evaluate value of function at given set of points: x 1 2 3 4…

Q: 29. Given a polynomial f(x) of degree n z 1, the fundamental theorem of algebra guarantees at least…

A:

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: 26. Determine whether the polynomial in Z[x] satisfies an Eisenstein Criterion for irreducibility…

A: see below the answer with explanation

Q: (1) Let ao = 0, a1 = 1, am = 3a,-1+ 10am-2 + 14 - 5" Find the solution of a, using the…

A: Given: a0 = 0 , a1 = 1 , an = 3an-1 +10 an-2 + 14·5n To determine: (a) The solution of an, using the…

Q: 7. Find a fourth-degree polynomial with integer coefficients that has zeros x=3i and x=-1. The root…

A:

Q: Suppose a functions satisfies f(5) = 2, f'(5) = 5, f"(5) = -7, and f"(5) = 12. What is the third…

A: We need to find correct option.

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: 1. The conventional algorithm for evaluating a polynomial a„x“ + an-1x" at x = c can be expressed in…

A:

Q: 6. Does there exist a polynomial of degree 4 with integer coefficients that has zeroes i, 2i, 3i and…

A: Complex root must be in conjugate pair

Q: In Horner's method suppose x0 is a root of p(x). show that the numbers bk(x0) produced by…

A: a. Let the polynomial be  Px=∑k=0nakxn-k                   (i) Given that x=x0 be a root of the…

Q: 7. Find a polynomial of degree 3 that has the given zeros: -1, 1, 3 and a constant term equal to -9.

A:

Q: The taylor polynomial of order 3 generated by f(x) - at a-0 81 x3 1 P = 3. 9. 27 81 x3 x4 6. 27 81…

A: Taylor polynomial

Q: Find the residue of 3 Z 2 z² - 1 at z→ ∞0.

A: We have to find the solution of the given problem.

Q: c) Prove that there cannot exist two different polynomials q,r E K[x], both of degree less than n,…

A:

Q: 1. Use the definition to find the first five nonzero terms of the Taylor polynomial, centered at c =…

A: To find the Taylor polynomial of function f(x) =1/x centered at c=1

Q: 3. A polynomial of degree n 21 has exactly a zero of multiplicity m is counted m times. zeros if

A: To fill in the blanks where statement is A polynomial of degree n≥1 has exactly ......... zeros if a…

Q: 1 Let f (x) 1 + 3x The coefficient of x8 in the 8th Taylor polynomial pg is . (Write your answer as…

A:

Q: 1. If P2(x) is the second degree polynomial that interpolates to f(x) = 1+x 0,0.1, 0.2, find a…

A: To find- If P2x is the second degree polynomial that interpolates to fx = 61 + x at x = 0, 0.1, 0.2,…

Q: #10. A polynomial f(x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: At the given condition:                Polynomial with real coefficients , each complex root…

Q: Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0

A: Given: To find their intersection using matrices. Given two polynomials in from of…

Q: 2.3 Theorem. Let p(x) be an irreducible polynomial in F[x). Then there exists an extension E of F in…

A:

Q: 1. For positive integers i, j,n such that i+ j = n (a) show that corresponding multinomial…

A: sol(a) we know that multinomial co-efficient is definedby ni,j=n!i!,j!  —(1) where i+j=nnow…

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: 5. (a) Show that x2 – 5 does not have a zero in Q[/2]. (b) Prove that Q[/2] is not isomorphic to…

A: Zeros of the polynomial x2 - 5 are only sqrt(5) and -sqrt(5) which doesn't belongs to Q [ sqrt(2) ]…

Q: 8. Compute the Taylor polynomial of degree 4 about a = 1 for f(r) = 1/x. Write out the 1st 4 nonzero…

A: Given: f(x)=1x a=1

Q: 2. Prove that a function which is analytic in the whole plane and satisfies an inequality |f(2)| <…

A: