Functions: math help!!!!?

Question

f(x) has a property that f(2-x) = f(2+x) for all x. If f(x) has exactly 4 real zeros, find their sum.

Please show me (step by step) how to do this problem.

Thanks...

Answer 1

f(x) has exactly 4 real zeros ==>

f(x)=x⁴+ax³+bx²+cx+d and

f(x)=(x-x1)(x-x2)(x-x3)(x-x4),

where x1, x2, x3 and x4 are the 4 real zeros of the function ==>

x1+x2+x3+x4=-a

f(2-x)=f(2+x) ==>

(2-x)⁴+a(2-x)³+b(2-x)²+c(2-x)+d=

(2+x)⁴+a(2+x)³+b(2+x)²+c(2+x)+d ... (1)

Simplifying (1) leads to:

2(8+a)x³+2(32+12a+4b+c)x=0 ==>

(8+a)x²+(32+12a+4b+c)=0 ==>

8+a=0 and 32+12a+4b+c=0 ==>

a=-8 ==> -a=8 ==> x1+x2+x3+x4=8