Question 2. Are the polynomials x? – 2x + 1, 2x2 – 3x – 6, a) Linearly independent? -x2 + 5x – 3, 3x² + 8x – 6 € P2: YES - NO

Q: Let F = Q, s(x) = x² + 1, and E = Q[r]/(x² + 1)Q[x]. i) Factor the polynomial r? +1€ Q[r] into…

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Q: 2.1 Find the appropriate Lagrange interpolating polynomial using the following data table 2 0 0.5 1…

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Q: Question 17 Consider a polynomial f (x) = ax³ – bx + c. Find the value of b if f' (2) = 0 and f" (2)…

A: To find out the value of b.

Q: Let f, g and h are quadratic polynomials such that f(x) fíx + 1)= g(h(x)). If a, B1 are roots of…

A: Given that f, g and h are quadratic polynomials such that fxfx+1=ghx. Since α1, β1 are roots of…

Q: O pts Question 6 Consider the polynomial p(æ) = 45x3 + 69x? – 24x – 48 What are the (real-valued)…

A: Given is the third degree polynomial:

Q: Question (a) Use the numbers (called nodes) xo = 2, x1 = 2.75, and x2 = 4 to find the second…

A: Given: x0 = 2, x1 = 2.75,  x2 = 4,  f(x) = 1x To find: (a) Using the given numbers to find the…

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

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Q: QUESTION 11 Express the polynomial x3+2x+3 EZ-[x]as a product of irreducible polynomials of Zs[x]:…

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Q: a) Show that the cubic polynomials P(x) = 3 − 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x − 1) and Q(x) =…

A: Given two cubic polynomial,

Q: a) Get the (2n+1)x Pn (x) = (n +1) Prar tx) +o Pau (x) reconnection polynomials using the; P(x, 2)=…

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Q: Let P(x) be the second degree Lagrange interpolating polynomial that fits the data (0,0), (1,2) and…

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Q: For which real values of k do the following polynomials form a linearly dependent set in P,. 2-2x+kx…

A: Explained below...

Q: Question 4 11:10:23 Find the value ofn EN such that the polynomial f(x) = (x + 1)" + (x – 1)" is…

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Q: Are the polynomials x? + 1,x? – 1, 2x – 1 linearly independent in P2?

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Q: Let (-1,(-1) = 2), (0, ƒ(0) = 5) and (1, f(1) = 6). Which one gives quadratic spline functions?…

A: Given, -1,f(-1)=2,(0,f(0)=5) and 1,f(1)=6

Q: 6. If T(x) property have these polynomials? (4x – 2) T - T. What orthogonality T. (2x - 1), then T =…

A: We have to find the inner product w.r.t which the polynomials Tn(x) are orthogonal.

Q: Find a 4th degree polynomial f(x) with integer coefficients, such that 2i, -1, and 2 are roots of…

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Q: 10.2. Let f(x) = 3x³ + x* + x³ + 3x? + 2x +4 and g(x) = 2x³ + 3x² + x +1 be polynomials in Zs[x].…

A: Given that f(x)=3x5+x4+x3+3x2+2x+4 , g(x)=2x3+3x2+x+1 To find q(x) ,r(x): Here Dividend is…

Q: I Example 6.10. Assuming that the following values of y belong to a polynomial of degree 4, compute…

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Q: Example 6.10. Assuming that the following values of y belong to a polynomial of degree 4, compute…

A: Given values of y are x 0 1 2 3 4 5 6 7 y 1 -1 1 -1 1                Let the value for x=5,…

Q: Consider the polynomials a(x) = r' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find…

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Q: Consider the homogenous polynomial ai + 10r1r2 + x². Find a suitable substitution s = Question 3 azi…

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Q: Let f(x) = , find L, (x) of the second Lagrange interpolating polynomial, if x, =2, x, =2.5 , x, =4…

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Q: Find a polynomial of degree 3 (here ao, a1, a2, a3 are real numbers) such that p(x) = ao + a₁x +…

A: This question can be solved using the concept of solving system of linear equation using matrix.

Q: (d) Use Lagrange Interpolation to find an interpolating polynomial of the form axt + bæ³ + ca? + dx…

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Q: Test the sets of polynomials for linear inde­pendence. For those that are linearly dependent,…

A: We have to determine whether or not we can find real numbers r, s,t, which are not all zero, Such…

Q: Question 6 Determine the Charactaristic polynomial of x?y" + x?y" - 2xy' + 2y 0 OA None of the…

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Q: Let P(x) be the quadratic (2nd degree) Lagrange interpolating polynomial at the interpolation points…

A: Suppose we are given three distinct points x1,y1,x2,y2 and  x3,y3 then the quadratic Lagrange's…

Q: Factor f (x) = 2x^3 - 3x^2 - 5x + 6 into linear factors given 1 is a zero of f(x).

A: Given that f(x) = 2x³ - 3x² - 5x + 6

Q: Question 5 Find a, b eR such that the polynomial f(x) = 6x – 7x + ax? + 3x + 2 is divisible by g(x)…

A: Use division algorithm and divide the polynomial f(x) with g(x).

Q: If a, Band y are the zeroes of the polynomial p (x) = ax'+ 3bx² + 3cx+ d and having relation 2B =…

A: Option (c) is correct.

Q: x3 + (2i – 1)x³ – 2ix – 5 a = 1- i -

A: Given: The given polynomial is x5+2i-1x3-2xi-5 a=1-i into binomial.

Q: 18. Find the decomposition of the function 8(x)%3D(2x²+0.5)* three different ways.

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Q: snip

A: Given{x,2x-x2,3x+2x2}To check dependentIf they are dependent , need to express  one function in…

Q: Let f(x) = , find L,(x) of the second Lagrange interpolating polynomial, if x, =2, x, =2.5 , x, =4…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Write the polynomial f(t) = af + bt +e as a linear combination of the polynomials p = (t – 1), P2 =t…

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Q: Let f(x) = , find L, (x) of the second Lagrange interpolating polynomial, if x, =2, x, =2.5, X2 =4…

A: Find second order Lagrange interpolation polynomial

Q: QUESTION 11 Express the polynomial x3 + 2x+3 EZ;[x]as a product of irreducible polynomials of Z5[x].…

A: Please note in Z5, -1=4.

Q: Which pairs of polynomials f, 9 € C[X] do have exactly one common root? O f- (X - 1), g = (X + X2+X+…

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Q: Which of the sets of polynomials is Linearly independent? O (1+x, 1-X) O (1, X, 0} O (1+X, 2+2X} O…

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Q: Question 2 Prove that the polynomial f(x) = x2 + a2", n € N, a E R\ {0} is not divisible by g(x) =…

A: Factor theorem: The polynomial f(x) has a factor x-k if and only if f(k)=0. Given: f(x)=x2n+a2n,…

Q: 2 17 1 12 7 15 2] What is the coefficient of X in the characteristic polynomial of the matrix A =…

A: Given matrix 5211127152Form a new matrix by subtracting λ from the diagonal…

Q: -2 21 -9-21 1- FIND THE INTER POLATION POLYNOMIAL BY USING THE FOR THE ABOVE NEWTON DIVIDED…

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Q: -4 -11 25 16. Without formally showing that there is a nontrivial linear combination of x-2x, x++x³…

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Q: Suppose that Pɔ(x) is the second degree Lagrange interpolating polynomial for the data: 1 2 Yi 3 6…

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Q: Question 2: For which real values of a do the polynomials P1(t) = at? -t -;; P2(t) = -;t² + at P3(t)…

A: In the given question we have to find the value of a for which the given three polynomials are…

Q: #37

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Q: In the set of polynomials in the form: P(x) = ax + bx' + cx² + dx + e, x E R there is a subset M,…

A: Given the polynomial M=px=ax4+bx3+cx2+dx+e, x∈ℝ|x,p(x)=-2,4,-1,0,1,-2,3,6. (a) Consider a cubic…

Q: A. Show that if p(n) is a polynomial of the form n3 + c2 n² + C1 n + co, then p(n) = ©(n³).

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Q: It is possible to write æ* as a linear combination of Chebyshev's polynomials, for example a² =…

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