In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 4x² − 9x + 36

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This process is essential for simplifying expressions and solving equations. Techniques such as grouping, using the distributive property, and applying special factoring formulas (like the difference of squares) are commonly employed.

Rational Root Theorem

The Rational Root Theorem provides a method for identifying possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is useful for testing potential roots to simplify the polynomial.

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 crucial in algebra, as it indicates that the polynomial does not have any rational roots or simpler factors, which can affect the methods used for solving equations.