36. (Remainder Theorem) Let f(x) = F[x] where F is a field, and let a € F. Show that the remainder r(x) when a, in accordance with the division algorithm, is f(a). f(x) is divided by x -
36. (Remainder Theorem) Let f(x) = F[x] where F is a field, and let a € F. Show that the remainder r(x) when a, in accordance with the division algorithm, is f(a). f(x) is divided by x -
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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![36. (Remainder Theorem) Let f(x) = F[x] where F is a field, and let a € F. Show that the remainder r(x) when
f(x) is divided by x - a, in accordance with the division algorithm, is f(a).
23.1 Theorem (Division Algorithm for F[x])
Let
f(x) = anx² + an-1x²-1
and
+
g(x) =
: bmxm + bm-1xm-1
+ ao
+
+ bo
be two elements of F[x], with an and bm both nonzero elements of F and m > 0. Then
there are unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x)+r(x),
where either r(x) = 0 or the degree of r(x) is less than the degree m of g(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ada9a20-cfc4-4295-96e5-fd4aa8bf1cec%2F6338f48c-b4fd-4602-b8be-4759b2df3fc8%2Ffj0k1um_processed.jpeg&w=3840&q=75)
Transcribed Image Text:36. (Remainder Theorem) Let f(x) = F[x] where F is a field, and let a € F. Show that the remainder r(x) when
f(x) is divided by x - a, in accordance with the division algorithm, is f(a).
23.1 Theorem (Division Algorithm for F[x])
Let
f(x) = anx² + an-1x²-1
and
+
g(x) =
: bmxm + bm-1xm-1
+ ao
+
+ bo
be two elements of F[x], with an and bm both nonzero elements of F and m > 0. Then
there are unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x)+r(x),
where either r(x) = 0 or the degree of r(x) is less than the degree m of g(x).
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