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Neat Fact for Integral Solutions to a polynomial

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Neat Fact for Integral Solutions to a polynomial [#permalink] New post 02 Jul 2013, 01:09
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Hello!

Consider any polynomial \(f(x) = A_1x^n+A_2x^{n-1}+.....A_n\)

Assumption : All the co-efficients for the given polynomial have to be integral,i.e. \(A_1,A_2,A_3....A_n\) are all integers.

Fact:Any integral solution(root) for the above polynomial will always be a factor(positive/negative) of the constant term : \(A_n\)

Example I : \(f(x) = 5x^2-16x+3\). Thus, we know that if the given polynomial has any integral solutions, then it will always be out of one of the following : \(-3,-1,1,3\)

We see that only x=3 is a root for the given polynomial. Also, we know that product of the roots is\(\frac {3}{5}\). Hence, the other root is \(\frac {1}{5}\)

Example II : Find the no of integral solutions for the expression \(f(x) = 3x^4-10x^2+7x+1\)

A. 0
B. 1
C. 2
D. 3
E. 4

For the given expression, instead of finding the possible integral solutions by hit and trial, we can be rest assured that if there is any integral solution, it will be a factor of the constant term ,i.e. 1 or -1. Just plug-in both the values, and we find that f(1) and f(-1) are both not equal to zero. Thus, there is NO integral solution possible for the given expression--> Option A.

Example III : Find the no of integral solutions for the expression \(f(x) = 4x^4-8x^3+9x-3\)

A. 0
B. 1
C. 2
D. 3
E. 4

Just as above, the integral roots of the given expression would be one of the following : -3,-1,1,3. We can easily see that only x = -1 satisfies. Thus, there is only one integral solution for the given polynomial-->Option B.

Hence, keeping this fact in mind might just reduce the range of the hit and trial values we end up considering.
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Hit and Trial for Integral Solutions

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Re: Neat Fact for Integral Solutions to a polynomial [#permalink] New post 17 Mar 2015, 09:58
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Re: Neat Fact for Integral Solutions to a polynomial   [#permalink] 17 Mar 2015, 09:58
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