Proving Irreducibles

Prove that the polynomial

$(x-a_1)(x-a_2)...(x-a_n)-1$ ,

where $a_1, a_2, ..., a_n$ are distinct integers, cannot be written as the product of two non-constant polynomials with integer coefficients, i.e., it is irreducible.

#Algebra

Easy Math Editor

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Comments

Let $f(x) = (x-a_1)(x-a_2)(x-a_3)..........(x-a_n) - 1$ For sake of contradiction let us consider $f(x)$ can be written as the product of two polynomials with integer co-efficients,ie, $f(x) = p(x)q(x)$ . For all i = 1,...,n we have $f(a_i) = p(a_i)q(a_i) = -1$ . Since $p(x)$ and $q(x)$ are polynomials with integers co-efficients. we must have $p(a_i) = 1$ and $q(a_i) = -1$ OR $p(a_i) = -1$ and $q(a_i) = 1$

In either case we have $p(a_i)+q(a_i) = 0$ . Hence The polynomial $p(x) + q(x)$ must have n distinct roots. But the degrees of both $p(x)$ and $q(x)$ are less than n and hence $p(x)+ q(x)$ cannot have n roots. Hence Contradiction.

Eddie The Head - 7 years, 2 months ago

An interesting note here: Dropping the "distinct integer" restriction on the problem invalidates the claim. One simple counterexample is $(x+2)(x+2)-1=(x+1)(x+3)$ .

Daniel Liu - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$