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ISI Entrance Examination
2015 ISI Entrance Examination
4
Show that a negative number r exists such that $p(r)=q(r)$
Show that a negative number r exists such that $p(r)=q(r)$
Source: ISI Entrance 2015
May 10, 2015
polynomial
algebra
Problem Statement
Let
p
(
x
)
=
x
7
+
x
6
+
b
5
x
5
+
⋯
+
b
0
p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0
p
(
x
)
=
x
7
+
x
6
+
b
5
x
5
+
⋯
+
b
0
and
q
(
x
)
=
x
5
+
c
4
x
4
+
⋯
+
c
0
q(x) = x^5 + c_4 x^4 + \cdots +c_0
q
(
x
)
=
x
5
+
c
4
x
4
+
⋯
+
c
0
. If
p
(
i
)
=
q
(
i
)
p(i)=q(i)
p
(
i
)
=
q
(
i
)
for
i
=
1
,
2
,
3
,
⋯
,
6
i=1,2,3,\cdots,6
i
=
1
,
2
,
3
,
⋯
,
6
. Show that there exists a negative integer r such that
p
(
r
)
=
q
(
r
)
p(r)=q(r)
p
(
r
)
=
q
(
r
)
.
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