Any help would be appreciated The Rational squareroot Theore

Any help would be appreciated!!

The Rational squareroot Theorem. Let p(x) = a_n x^n + a_n-1 x^n-1 +... + a_0 element of Z[x], where a_n notequalto 0. Prove that if p(r/s) = 0, where gcd(r, s) = 1, then r|a_0 and s| a_n.

Solution

Let p(x) = a_n xⁿ + a_n1xⁿ¹ + ... + a₁x + a₀ for some a₀, ..., a_n Z, and suppose p(r/s) = 0 for some co- prime r, s Z:

p (r/s) = a_n ( r/s)ⁿ + a_n-1 ( r/s)^n-1 +… + a₁ (r/s) + a₀ = 0

If we multiply both sides by sⁿ, shift the constant term to the right hand side, and factor out r on the left hand side, we get, r (a_n r^n-1 + a_n-1 sr^n-2 +… + a₁s^n-1) = - a₀sⁿ.

We see that r times the integer quantity in parentheses equals - a₀sⁿ , so that r divides a₀sⁿ But r is co-prime to s and therefore to sⁿ, so by (the generalized form of) Euclid\'s lemma it must divide the remaining factor a₀ of the product.

If we, instead, shift the leading term to the right hand side and factor out s on the left hand side, we get

s( a_n-1r^n-1 + a_n-2sr^n-2 + …+ a₀ s^n-1 ) = -a_n rⁿ

And for similar reasons, we can conclude that s divides a_n