Consider the following data set Xi -2 0 12 f(x₁) 5 1 -1 5 (a) Construct the quadratic Lagrange interpolating polynomial P₂(x) that interpo- lates f(x) at x₁ for i = 0,1,2. Sketch the three Lagrange basis polynomials on [To, x₂]. (b) Construct the cubic Lagrange interpolating polynomial P3(r) that interpolates f(x) at x; for i = 0, 1, 2, 3. Sketch the four Lagrange basis polynomials on [10, 13)

Q: 2. Find the polynomial the passes through the following points (0,0), (2, —2), (4,0)

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Q: Find a polynomial y = p(x) of degree 4 that goes through the points (0, 4), (1, 2), (2, -4), (3,…

A: solution of the given question is like this:

Q: Show that Q(4 - i) = Q(1 + i), where i = √-1.

A: To show that ℚ4-i=ℚ1+i. That is we need to show that ℚ4-i⊆ℚ1+i and ℚ1+i⊆ℚ4-i. Here i=-1 that is…

Q: Let a0, a1, a2, a3, a4 be constant real numbers such thata0/5 + a1/4 + a2/3 + a3/2 + a4 = 0.Show…

A: Consider−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−f(x)=5a0​x5​+4a1​x4​+3a2​x3​+2a3​x2​+a4​x.with  input from…

Q: Suppose that the functions s and t are defined for all real numbers x as follows. s(x)=2x+5 t(x)=3x…

A: Step 1: Given functions: s(x)=2x+5, t(x)=3xwe have to find the expressions: (s⋅t)(x), (s+t)(x), and…

Q: Find the unique quadratic polynomial y = ax² +bx+c that passes through the three points (1, 1),…

A: What is Polynomial: A polynomial is a real valued function. It is an algebraic function that is…

Q: ve that if x5 – 3x4 + 2x

A: Introduction: In an equation, the two expressions are assumed to be equal, as indicated by the…

Q: 6. (10pts) Determine whether you can find the interpolating polynomial p(t) = a, + a,t + a,r² for…

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Q: Define the Newton form of the interpolating polynomial?

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Q: Calculate the 1st order Newton polynomial that interpolates the two data points (1,1),(2,3). Now…

A: We can use Newton's forward difference interpolation method to find Newton's polynomial.

Q: Let consider given points: (0, 1), (2,3), (3,0). (a) Use Lagrange interpolation to find a polynomial…

A: Note: “Since you have posted multiple questions, we will provide the solution only to the first…

Q: (b) Are the polynomials x + 5, x – 5, 2x +1 linearly independent in P1(R)? Explain.

A: The vectors a1, ..., an are called linearly independent if there are no non-trivial combination of…