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2003 AIME Problems
9
Two Polynomials
Two Polynomials
Source:
December 26, 2006
algebra
polynomial
Problem Statement
Consider the polynomials
P
(
x
)
=
x
6
−
x
5
−
x
3
−
x
2
−
x
P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x
P
(
x
)
=
x
6
−
x
5
−
x
3
−
x
2
−
x
and
Q
(
x
)
=
x
4
−
x
3
−
x
2
−
1.
Q(x)=x^{4}-x^{3}-x^{2}-1.
Q
(
x
)
=
x
4
−
x
3
−
x
2
−
1.
Given that
z
1
,
z
2
,
z
3
,
z_{1},z_{2},z_{3},
z
1
,
z
2
,
z
3
,
and
z
4
z_{4}
z
4
are the roots of
Q
(
x
)
=
0
,
Q(x)=0,
Q
(
x
)
=
0
,
find
P
(
z
1
)
+
P
(
z
2
)
+
P
(
z
3
)
+
P
(
z
4
)
.
P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).
P
(
z
1
)
+
P
(
z
2
)
+
P
(
z
3
)
+
P
(
z
4
)
.
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