f(x) is a 6th degree polynomial satisfying f(1-x)=f(x+1) for all real values of x. If f(x) has four distinct real roots and two real and equal roots, then find the sum of all roots of f(x)=0.
The function is symmetrical about the line x=1. For example f(2) = f(0), f(5) = f(-3), f(1) = f(1). What confuses things is the use of x in the condition: when that x is zero the argument to f is 1, which is traditionally the x value; it's a different x.
Since it's symmetric about x=1, first of all the two equal roots are at x=1. As far as the total goes, the easiest way to that is via the average. The average of the roots must be 1. As there are 5 roots, the total is 5. If you were to count the double root twice, the total would be 6.
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Posted by Charlie
on 2019-09-09 09:29:05 |
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