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a. The polynomials P₁(x)=x¹, P₂(x)=x² and P3(x)=x³ form a basis of the space P3 of polynomials of degree at most 3. Ob. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁

a. The polynomials P₁(x)=x¹, P₂(x)=x² and P3(x)=x³ form a basis of the space P3 of polynomials of degree at most 3. Ob. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Which of the following statements are true?

 

a. The polynomials P₁(x)=x¹, P₂(x)=x² and P3(x)=x³ form a basis of the space P3 of polynomials of degree at most 3. b. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁<x1<X2. O c. We have dim(Pn)=n, where P₁ denotes the space of all polynomials of degree at most n. d. Given pairwise distinct nodes X0,...,Xn‚Xn+1ER, the associated Newton polynomials satisfy the identity Nn+1(x)=(x-xn)Nn(x) for all XER.
Transcribed Image Text:a. The polynomials P₁(x)=x¹, P₂(x)=x² and P3(x)=x³ form a basis of the space P3 of polynomials of degree at most 3. b. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁<x1<X2. O c. We have dim(Pn)=n, where P₁ denotes the space of all polynomials of degree at most n. d. Given pairwise distinct nodes X0,...,Xn‚Xn+1ER, the associated Newton polynomials satisfy the identity Nn+1(x)=(x-xn)Nn(x) for all XER.
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