Question 32879: Let f(x) =x^n + a(sub1)x^n-1 + a(sub2)x^n-2..........a(sub n-1)x+a(sub n) be a polynomial with integer coefficients. Suppose there are four distinct integers a,b,c and d such that f(a)=f(b)=f(c)=f(d)=8. Can there be an integer k such that f(k)=3.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Since f(a)=f(b)=f(c)=f(d)=8, assume
f(x) = g(x) (x-a)(x-b)(x-c)(x-d) + 8, where g(x) is monic poly. in Z[x]
If f(k) = 3 , then g(k) (k-a)(k-b)(k-c)(k-d) = -5.
But 5 is a prime and g(k) is nan integer, a,b,c,d are distict.
Hence, at least two among (k-a),(k-b),(k-c),(k-d) would be equal to
1 or -1 , this is impossible.
Kenny
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