(1)
(a) Let
f
(
x
)
, g
(
x
) be polynomials with real coefficients.
Prove that there exists a
unique
polynomial
h
(
x
) =
x
k
+
a
k

1
x
k

1
+
. . .
+
a
0
such that
h
(
x
) divides both
f
(
x
) and
g
(
x
) and every other polynomial that divides both
f
(
x
) and
g
(
x
) divides
h
(
x
).
h
(
x
) is called the greatest common
divisor of
f
(
x
) and
g
(
x
) and is denoted by (
f
(
x
)
, g
(
x
)).
Hint:
Use Euclidean algorithm to construct
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 Spring '10
 Applebaugh
 Math, Polynomials, Greatest common divisor, real coefficients, Greatest common divisor of two polynomials

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