d. Prove that the ring of polynomial R[X] is also an integral domain. e. In Z₁[X], show that (2x+1)² = 1. Show that x = f(X)g(X) for some noncon- stant polynomials f(X) and g(X) in Z₁[X]. f. Does item 2. hold if R is not an integral domain.
d. Prove that the ring of polynomial R[X] is also an integral domain. e. In Z₁[X], show that (2x+1)² = 1. Show that x = f(X)g(X) for some noncon- stant polynomials f(X) and g(X) in Z₁[X]. f. Does item 2. hold if R is not an integral domain.
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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Question
answer letters d to f
![2. Let D be an integral domain and let D[X] denote the ring of polynomials in the
indeterminate X and coefficients coming from D. For a nonzero polynomial
f(x) € D[X], let deg f(X) denote the degree of the the polynomial f(x).
a. Prove that the constant term of the product f(X)g(X) is the product of
the constant terms of f(X) and g(X).
b. Prove that the leading coefficient of the product ƒ(X)g(X) is the product
of the leading coefficients of f(X) and g(X).
c. Prove that deg f(X)(g(X) = deg f(X) + deg g(X).
d. Prove that the ring of polynomial R[X] is also an integral domain.
e. In Z₁ [X], show that (2x + 1)² = 1. Show that x = f(X)g(X) for some noncon-
stant polynomials f(X) and g(X) in Z₁[X].
f. Does item 2. hold if R is not an integral domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e4a000c-d4c6-4ce2-961c-ef465030fee1%2Ff4f2e494-9c68-4ac8-8491-1569eab2390d%2Fumiw6b_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let D be an integral domain and let D[X] denote the ring of polynomials in the
indeterminate X and coefficients coming from D. For a nonzero polynomial
f(x) € D[X], let deg f(X) denote the degree of the the polynomial f(x).
a. Prove that the constant term of the product f(X)g(X) is the product of
the constant terms of f(X) and g(X).
b. Prove that the leading coefficient of the product ƒ(X)g(X) is the product
of the leading coefficients of f(X) and g(X).
c. Prove that deg f(X)(g(X) = deg f(X) + deg g(X).
d. Prove that the ring of polynomial R[X] is also an integral domain.
e. In Z₁ [X], show that (2x + 1)² = 1. Show that x = f(X)g(X) for some noncon-
stant polynomials f(X) and g(X) in Z₁[X].
f. Does item 2. hold if R is not an integral domain.
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