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4
c(x-a)(x-b)/(c-a)(cb)+a(x-b)(x-c)/(a-b)(a-c)+b(x-c)(x-a)/(b-c)(b-a)+1 HOMC17.4
c(x-a)(x-b)/(c-a)(cb)+a(x-b)(x-c)/(a-b)(a-c)+b(x-c)(x-a)/(b-c)(b-a)+1 HOMC17.4
Source:
September 9, 2019
algebra
polynomial
Problem Statement
Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial
P
(
x
)
=
c
(
x
−
a
)
(
x
−
b
)
(
c
−
a
)
(
c
−
b
)
+
a
(
x
−
b
)
(
x
−
c
)
(
a
−
b
)
(
a
−
c
)
+
b
(
x
−
c
)
(
x
−
a
)
(
b
−
c
)
(
b
−
a
)
+
1
P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1
P
(
x
)
=
(
c
−
a
)
(
c
−
b
)
c
(
x
−
a
)
(
x
−
b
)
+
(
a
−
b
)
(
a
−
c
)
a
(
x
−
b
)
(
x
−
c
)
+
(
b
−
c
)
(
b
−
a
)
b
(
x
−
c
)
(
x
−
a
)
+
1
. The value of
P
(
2017
)
P (2017)
P
(
2017
)
is(A):
2015
2015
2015
(B):
2016
2016
2016
(C):
2017
2017
2017
(D):
2018
2018
2018
(E): None of the above.
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