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 f(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).
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 f(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).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.3: Change Of Basis
Problem 17EQ
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![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%2F221a2d86-de3b-4a4c-9441-2b30b27b1bf2%2F0u3wsgp_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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