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Chennai Mathematical Institute B.Sc. Entrance Exam
2023 CMI B.Sc. Entrance Exam
3
3
Part of
2023 CMI B.Sc. Entrance Exam
Problems
(1)
If r is a double root of a degree 4 polynomial, then results on its coefficients
Source: CMI 2023 B3
5/9/2023
Consider the polynomial
p
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
p(x) = x^4 + ax^3 + bx^2 + cx + d
p
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
. It is given that
p
(
x
)
p(x)
p
(
x
)
has its only root at
x
=
r
x = r
x
=
r
i.e
p
(
r
)
=
0
p(r) = 0
p
(
r
)
=
0
.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
a
n
>
<span class='latex-bold'>(a)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
an
>
Show that if
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are rational then
r
r
r
is rational.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
a
n
>
<span class='latex-bold'>(b)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
an
>
Show that if
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers then
r
r
r
is an integer. (Hint: Consider the roots of
p
′
(
x
)
p'(x)
p
′
(
x
)
)
algebra
polynomial
CMI