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10
2010 Algebra #10: Functional Compositional Equation
2010 Algebra #10: Functional Compositional Equation
Source:
July 15, 2012
algebra
polynomial
Problem Statement
Let
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
be two cubic polynomials such that
p
(
0
)
=
ā
24
p(0)=-24
p
(
0
)
=
ā
24
,
q
(
0
)
=
30
q(0)=30
q
(
0
)
=
30
, and
p
(
q
(
x
)
)
=
q
(
p
(
x
)
)
p(q(x))=q(p(x))
p
(
q
(
x
))
=
q
(
p
(
x
))
for all real numbers
x
x
x
. Find the ordered pair
(
p
(
3
)
,
q
(
6
)
)
(p(3),q(6))
(
p
(
3
)
,
q
(
6
))
.
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