Question

a-b+c+d If a, b, c, d are distinct integer such that (x- a) (x-b) (x- c)(x-d) = 4 has an integer solution x= r. Find the ratio of
$\frac{a + b + c + d}{r}$
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Answer 1

Answer:

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Explanation:

Let a,b,c,d be distinct integers such that the equation (x−a)(x−b)(x−c)(x−d)−9=0 has an integer root r,then find the value of a+b+c+d−4r.

As r is the integer root of the equation (x−a)(x−b)(x−c)(x−d)−9=0,so (r−a)(r−b)(r−c)(r−d)=9

or (a−r)(b−r)(c−r)(d−r)=9

We need to find (a−r)+(b−r)+(c−r)+(d−r) but i do not know how to find that.

a,b,c and d are distinct integers.

So r−a,r−b,r−c and r−d are all distinct integers.

Again, 9=32

Hence the only possible factorisation:

(r−a)(r−b)(r−c)(r−d)=3⋅3⋅1⋅1

or

(a−r)(r−b)(c−r)(r−d)=3⋅3⋅1⋅1

So we have a=r+3, b=r−3, c=r+1, d=r−1.

Thus a+b+c+d−4r=0