Factor completely, or state that the polynomial is prime. x^2-11x+28

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler components, or factors. For quadratic polynomials of the form ax^2 + bx + c, this often means finding two binomials that multiply to give the original polynomial. Understanding how to identify these factors is crucial for simplifying expressions and solving equations.

Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation. This formula is particularly useful when factoring is difficult or when determining if a polynomial is prime. The discriminant (b² - 4ac) indicates the nature of the roots, which can help in understanding whether the polynomial can be factored.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is essential in algebra, as it helps in determining the limits of simplification and the methods needed for solving equations. In the context of quadratics, if the discriminant is negative, the polynomial is prime over the reals.