To submit Let R[] be the set of all expressions a = ao+ajx+a2x²+ · where aj e R for all nonnegative integers i. Informally, an element of R[] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R]. analogous to the definitions for R(x] in the notes. Given a,be R[x], your defi- nitions should indicate what each coefficient of the sum a +b and product ab is. (b) Let f = ao +a,x++a," be a polynomial. I can treat f as an element of R[[x]] by defining a,41,an+2,... all to equal 0. This shows that R[x] CR[k]. If you had already proved that R] was a ring, how could you use this fact to help you prove R[x] is a ringa (c) Let a E R[x]] with ao #0. Prove that a has a multiplicative inverse in R[x]. You may assume that the multiplicative identity element in R[x] is IRI] = 1+0x +0x²+0x+ ..., and that multiplication in R[] is commutative (Hint. If ab = 1Rj, equate coefficients and solve for bo, b1,b2,... in turn.]
To submit Let R[] be the set of all expressions a = ao+ajx+a2x²+ · where aj e R for all nonnegative integers i. Informally, an element of R[] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R]. analogous to the definitions for R(x] in the notes. Given a,be R[x], your defi- nitions should indicate what each coefficient of the sum a +b and product ab is. (b) Let f = ao +a,x++a," be a polynomial. I can treat f as an element of R[[x]] by defining a,41,an+2,... all to equal 0. This shows that R[x] CR[k]. If you had already proved that R] was a ring, how could you use this fact to help you prove R[x] is a ringa (c) Let a E R[x]] with ao #0. Prove that a has a multiplicative inverse in R[x]. You may assume that the multiplicative identity element in R[x] is IRI] = 1+0x +0x²+0x+ ..., and that multiplication in R[] is commutative (Hint. If ab = 1Rj, equate coefficients and solve for bo, b1,b2,... in turn.]
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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![To submit Let R[] be the set of all expressions
a = ao+ajx+azx²+...=Lax
i=0
where a; eR for all nonnegative integers i. Informally, an element of R[x] is like a
polynomial except that it can have infinitely many terms.
(a) Carefully write down definitions of addition and multiplication operations for R[x],
analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi-
nitions should indicate what each coefficient of the sum a +b and product ab is.
(b) Let f = ao +a,x+.+a,x" be a polynomial. I can treat f as an element of R[[x]
by defining an+1,an+2,... all to equal 0. This shows that R[x] CR[x].
If you had already proved that R[] was a ring, how could you use this fact to
help you prove R[x] is a ring
(c) Let a e R[x] with ao #0. Prove that a has a multiplicative inverse in R[x]. You
may assume that the multiplicative identity element in R[x] is
Ig = 1+0x+0x? +Qr³ + ...,
and that multiplication in R[x] is commutative
[Hint. If ab = 1RI], equate coefficients and solve for bo, b1,b2,... in turn.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b4d98a6-2ed4-451a-82b3-b3630dcf9fce%2F46b2721f-3d4e-4c25-b697-9381e63f1903%2F9lrxzdn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To submit Let R[] be the set of all expressions
a = ao+ajx+azx²+...=Lax
i=0
where a; eR for all nonnegative integers i. Informally, an element of R[x] is like a
polynomial except that it can have infinitely many terms.
(a) Carefully write down definitions of addition and multiplication operations for R[x],
analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi-
nitions should indicate what each coefficient of the sum a +b and product ab is.
(b) Let f = ao +a,x+.+a,x" be a polynomial. I can treat f as an element of R[[x]
by defining an+1,an+2,... all to equal 0. This shows that R[x] CR[x].
If you had already proved that R[] was a ring, how could you use this fact to
help you prove R[x] is a ring
(c) Let a e R[x] with ao #0. Prove that a has a multiplicative inverse in R[x]. You
may assume that the multiplicative identity element in R[x] is
Ig = 1+0x+0x? +Qr³ + ...,
and that multiplication in R[x] is commutative
[Hint. If ab = 1RI], equate coefficients and solve for bo, b1,b2,... in turn.]
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