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(a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m are nonnegative integers and ao,..., an, bo, ..., bm are real numbers such that an, bm # 0 and т consta, · idk E consto, · idk i=0 i=0 as relations, then n = m and a; = b; for all i = 0, ... , n. Hint: Induction on n starting at 0. 33 (b) If m1, m2, b1, b2 are real numbers, express lm2,b2 o lm1,b, as an affine function. (c) If m1, m2, b1, b2 are real numbers, express lm2,b2 + lmı,b1 as an affine function. (d) If m1, m2, b1, b2 are real numbers, express lm2,b2 · Imı,b1 as a quadratic function. (e) Demonstrate, if n and m are a1,..., an and b1,..., bm are real numbers, then ) ( m n+m k E consta, · idk E constb; · idg ) = E consta;br-i ) · id i=0 i=0 k=0 i=0 by induction on n. This shows the degree of the product of two polynomials is the sum of their degrees. (f) Demonstrate by way of example. The degree of the sum of two polynomials can be equal to the maximum of their degrees. (g) Demonstrate by way of example. The degree of the sum of two polynomials can be less than the maximum of their degrees. (h) Provide an example of two degree three polynomials whose sum has degree 0.

(a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m are nonnegative integers and ao,..., an, bo, ..., bm are real numbers such that an, bm # 0 and т consta, · idk E consto, · idk i=0 i=0 as relations, then n = m and a; = b; for all i = 0, ... , n. Hint: Induction on n starting at 0. 33 (b) If m1, m2, b1, b2 are real numbers, express lm2,b2 o lm1,b, as an affine function. (c) If m1, m2, b1, b2 are real numbers, express lm2,b2 + lmı,b1 as an affine function. (d) If m1, m2, b1, b2 are real numbers, express lm2,b2 · Imı,b1 as a quadratic function. (e) Demonstrate, if n and m are a1,..., an and b1,..., bm are real numbers, then ) ( m n+m k E consta, · idk E constb; · idg ) = E consta;br-i ) · id i=0 i=0 k=0 i=0 by induction on n. This shows the degree of the product of two polynomials is the sum of their degrees. (f) Demonstrate by way of example. The degree of the sum of two polynomials can be equal to the maximum of their degrees. (g) Demonstrate by way of example. The degree of the sum of two polynomials can be less than the maximum of their degrees. (h) Provide an example of two degree three polynomials whose sum has degree 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 2.38. (a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m are nonnegative integers and ao,..., an, bo, .. , bm are real numbers such that an, bm + 0 and n consta; · idg =Econsto; · idk i=0 i=0 as relations, then n = m and a; = b; for all i = 0, ..., n. Hint: Induction on n starting at 0. 33 (b) If m1, m2, b1, b2 are real numbers, express lm,b2 o lm1,b1 as an affine function. (c) If m1, m2, b1, b2 are real numbers, express lm2.b2 + lm1.b1 as an affine function. (d) If m1, m2, b1, b2 are real numbers, express lIm2,62 · lmı,b1 as a quadratic function. (e) Demonstrate, if n and m are a1, ..., an and b1, ..., bm are real numbers, then ) (E m n+m k consta; · idg consti, idg ) = £E consta;bk-i id i=0 i=0 k=0 i=0 by induction on n. This shows the degree of the product of two polynomials is the sum of their degrees. (f) Demonstrate by way of example. The degree of the sum of two polynomials can be equal to the maximum of their degrees. (g) Demonstrate by way of example. The degree of the sum of two polynomials can be less than the maximum of their degrees. (h) Provide an example of two degree three polynomials whose sum has degree 0.
Transcribed Image Text:Exercise 2.38. (a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m are nonnegative integers and ao,..., an, bo, .. , bm are real numbers such that an, bm + 0 and n consta; · idg =Econsto; · idk i=0 i=0 as relations, then n = m and a; = b; for all i = 0, ..., n. Hint: Induction on n starting at 0. 33 (b) If m1, m2, b1, b2 are real numbers, express lm,b2 o lm1,b1 as an affine function. (c) If m1, m2, b1, b2 are real numbers, express lm2.b2 + lm1.b1 as an affine function. (d) If m1, m2, b1, b2 are real numbers, express lIm2,62 · lmı,b1 as a quadratic function. (e) Demonstrate, if n and m are a1, ..., an and b1, ..., bm are real numbers, then ) (E m n+m k consta; · idg consti, idg ) = £E consta;bk-i id i=0 i=0 k=0 i=0 by induction on n. This shows the degree of the product of two polynomials is the sum of their degrees. (f) Demonstrate by way of example. The degree of the sum of two polynomials can be equal to the maximum of their degrees. (g) Demonstrate by way of example. The degree of the sum of two polynomials can be less than the maximum of their degrees. (h) Provide an example of two degree three polynomials whose sum has degree 0.
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