document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #19326 by khwang(438) $\"\"$ $\"About$

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Since f(a)=f(b)=f(c)=f(d)=8, assume
\n" ); document.write( " f(x) = g(x) (x-a)(x-b)(x-c)(x-d) + 8, where g(x) is monic poly. in Z[x]\r
\n" ); document.write( "\n" ); document.write( " If f(k) = 3 , then g(k) (k-a)(k-b)(k-c)(k-d) = -5.
\n" ); document.write( " But 5 is a prime and g(k) is nan integer, a,b,c,d are distict.
\n" ); document.write( " Hence, at least two among (k-a),(k-b),(k-c),(k-d) would be equal to
\n" ); document.write( " 1 or -1 , this is impossible.\r
\n" ); document.write( "\n" ); document.write( " Kenny \n" ); document.write( "

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