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Greece Junior Math Olympiad
1988 Greece Junior Math Olympiad
3
factor polynomials - Greece Juniors 1988 p3
factor polynomials - Greece Juniors 1988 p3
Source:
September 13, 2024
algebra
polynomial
Problem Statement
Consider the polynomials
P
(
x
)
=
x
4
−
3
x
3
+
x
−
3
,
Q
(
x
)
=
x
2
−
2
x
−
3
R
(
x
)
=
−
x
2
−
5
x
+
a
P(x)=x^4-3x^3+x-3,\,\,\,\,Q(x)=x^2-2x-3 \,\,\,\, R(x)=-x^2-5x+a
P
(
x
)
=
x
4
−
3
x
3
+
x
−
3
,
Q
(
x
)
=
x
2
−
2
x
−
3
R
(
x
)
=
−
x
2
−
5
x
+
a
i) Find
a
∈
a \in
a
∈
R such that polynomial
R
(
x
)
R(x)
R
(
x
)
is dividide by
x
−
2
x-2
x
−
2
ii) Factor polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
iii) Prove that exrpession
−
x
2
+
x
+
P
(
x
)
Q
(
x
)
+
15
-x^2+x+\frac{P(x)}{Q(x)}+15
−
x
2
+
x
+
Q
(
x
)
P
(
x
)
+
15
is a perfect square.
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