Which polynomials are prime? Choose two correct answers.
A polynomial is considered prime (or irreducible) over the rational numbers \(\mathbb{Q}\) if it cannot be factored into the product of two or more non-constant polynomials with rational coefficients. For a quadratic polynomial \(ax^2 + bx + c\), we can determine its irreducibility by examining its discriminant \(D = b^2 - 4ac\). A quadratic polynomial is reducible over \(\mathbb{Q}\) if and only if it has rational roots, which occurs when the discriminant \(D\) is a perfect square of a rational number. If the discriminant is negative, the polynomial has no real roots and is therefore irreducible. If the discriminant is positive but not a perfect square, the roots are irrational, and the polynomial is also irreducible over \(\mathbb{Q}\).
We test each candidate polynomial:
Based on our analysis, the prime (irreducible) polynomials among the given choices are \(x^2 + 9\) and \(x^2 + 3x + 9\). Both have negative discriminants, preventing them from having any real or rational roots. The other two, \(x^2 - 9\) and \(-2x^2 + 8\), were shown to be factorable. Therefore, the correct choices are \(\boxed{x^2 + 9}\) and \(\boxed{x^2 + 3x + 9}\).
Identifying irreducible polynomials is a foundational skill in algebra, particularly when working with partial fraction decomposition or simplifying complex expressions. For example, in electrical engineering, when analyzing systems represented by transfer functions (rational expressions of polynomials), one must often factor the denominator to find the system's poles. Knowing whether a quadratic component can be broken down further or if it represents an irreducible quadratic factor (which corresponds to oscillatory, non-decaying components in time-domain analysis) is crucial for accurate system modeling and control.