Question
Question: If f(x) = | x | x-3 | 2x²-18 | 3x³-81 | | --- | --- | ------- | -------- | | 1 | x-5 | 2x²-50 ...
If f(x) =
| x | x-3 | 2x²-18 | 3x³-81 |
|---|---|---|---|
| 1 | x-5 | 2x²-50 | 4x³-500 |
| 2 | |||
| 3 |
then f(1) f(3)+f(3) f(5)+f(5) f(1) is equal to
A
f(1)+f(5)
B
f(3)+f(5)
C
f(1)+f(3)
D
f(1)+f(3)+f(5)
Answer
f(1)+f(3)
Explanation
Solution
By writing the cubic f(x) in two bases (one expanded about x=3 and the other about x=5) one finds that the “cross–product” combination f(1)f(3)+f(3)f(5)+f(5)f(1) simplifies exactly to f(1)+f(3).