Let f(x)=x^2+100 x+n (a) Give an infinite set of integers n...

Let $f(x)=x^{2}+100 x+n$
(a) Give an infinite set of integers $n$ such that $f(x)$ is reducible over $\mathbf{Q}$.
(b) Give an infinite set of integers $n$ such that $f(x)$ is irreducible over $\mathbf{Q}$.

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Key Concepts

Quadratic Discriminant

For any quadratic polynomial ax² + bx + c, the discriminant is the expression b² - 4ac. The value of the discriminant reveals the nature of the roots. When the discriminant is a perfect square and particularly nonnegative, the roots are rational (if a, b, c are rational), indicating that the quadratic is reducible over Q. If it is not a perfect square, then the quadratic is irreducible over Q.

Infinite Family Construction

This concept refers to the method of parametrically defining a set of integers (or other numbers) such that a condition holds for an infinite number of choices. In the context of polynomial reducibility or irreducibility, one can express the constant term in terms of another parameter (often related to the discriminant) to produce infinitely many instances that satisfy the desired factorization properties.

Polynomial Reducibility over Q

This concept involves determining whether a polynomial with rational coefficients can be factored into lower?degree polynomials that also have rational coefficients. In the case of quadratic polynomials, reducibility over Q happens if and only if the quadratic has two rational roots, which means its discriminant is a perfect square in Q.

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