Let f(x)=am x^m+am-1 x^m-1+⋯+a0 and g(x)=bn x^n+bn-1 x^n-1+ ⋯+b0 belong to Q[x] and suppose that

step 1 Let us consider the f of x a x square plus b x plus c is an integer whenever x is an integer. Fi

Let f(x)=am x^m+am-1 x^m-1+⋯+a0 and g(x)=bn x^n+bn-1 x^n-1+...

Let $f(x)=a_{m} x^{m}+a_{m-1} x^{m-1}+\cdots+a_{0}$ and $g(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+$
$\cdots+b_{0}$ belong to $Q[x]$ and suppose that $f(x) g(x)$ belongs to $Z[x] .$ Prove that $a_{i} b_{j}$ is an integer for every $i$ and $j .$

Key Concepts

Clearing Denominators

This concept involves multiplying a polynomial with rational coefficients by a common multiple of the denominators in order to convert it into a polynomial with integer coefficients. By doing so, one can work in the integral domain, which helps in applying integer divisibility arguments to the coefficients of the original rational polynomials.

Polynomial Multiplication and Coefficient Convolution

When multiplying two polynomials, each coefficient in the product is obtained as a sum of products (a convolution) of coefficients from the individual factors. Recognizing that the overall sum is an integer because the product polynomial has integer coefficients forces conditions on the individual products of coefficients, leading to insights about the integrality of each product.

Gauss's Lemma and Primitive Polynomials

Gauss's lemma states that the product of two primitive polynomials (polynomials with integer coefficients that have no common factor other than ±1) is also primitive. In problems where polynomials with rational coefficients are transformed into polynomials with integer coefficients via clearing denominators, Gauss's lemma can be applied to reason about the integrality and divisibility properties of their individual factors.

Rational versus Integral Coefficients

This concept emphasizes the difference between rational and integer coefficients in polynomials. When the product of two polynomials with rational coefficients yields a polynomial with integer coefficients, there is an underlying structural restriction. Specifically, the cancellation of denominators implies that products of individual coefficients must align in such a way that, when combined appropriately, they produce integer values.

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