Find a polynomial with all integral coefficients between 0 and 9, inclusive, such that P(−2) = P(−5) = 2006.
P(x)=
1x^11+6x^10+6x^9+6x^8+5x^7+1x^5+9x^4+4x^3+6
I started by finding a simpler polynomial, f, such that f(-5)=2006
By fiddling around with a spreadsheet I was able to concoct
f(x)=4x^4+4x^3+6.
However
f(-2)=38
The key was to realize (-5)^n+5*(-5)^(n-1)=0,
this allows adding more terms to f(x) which will not change f(-5) but will change f(-2)
(-2)^n+5(-2)^(n-1)=3(-2)^(n-1)
f(-2)=38, which is 1968 = 3*656 too small
I was able to add pairs of 1 and 5 coefficients to adjacent terms until I got a perfect match.
https://docs.google.com/spreadsheets/d/1_sOhOpK1bvpCO_zE_aXtlzrHXFTw-t7h8l764CF-Qnw/edit?usp=sharing
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Posted by Jer
on 2024-11-17 18:57:50 |
Solution