MathDB
Problems
Contests
Undergraduate contests
VTRMC
1980 VTRMC
4
4
Part of
1980 VTRMC
Problems
(1)
1980 VTRMC #4
Source:
8/11/2020
Let
P
(
x
)
P(x)
P
(
x
)
be any polynomial of degree at most
3.
3.
3.
It can be shown that there are numbers
x
1
x_1
x
1
and
x
2
x_2
x
2
such that
∫
−
1
1
P
(
x
)
d
x
=
P
(
x
1
)
+
P
(
x
2
)
,
\textstyle\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),
∫
−
1
1
P
(
x
)
d
x
=
P
(
x
1
)
+
P
(
x
2
)
,
where
x
1
x_1
x
1
and
x
2
x_2
x
2
are independent of the polynomial
P
.
P.
P
.
(a) Show that
x
1
=
−
x
2
.
x_1=-x_2.
x
1
=
−
x
2
.
(b) Find
x
1
x_1
x
1
and
x
2
.
x_2.
x
2
.
algebra
polynomial