Factor each polynomial. See Examples 5 and 6. 27-r^3

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and understanding polynomial behavior. Common methods include factoring out the greatest common factor, using special products, and applying techniques like grouping or the quadratic formula.

Difference of Cubes

The expression 27 - r^3 is a difference of cubes, which can be factored using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, 27 is 3^3 and r^3 is r^3, allowing us to identify a = 3 and b = r. Recognizing this pattern is crucial for efficiently factoring such expressions.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the expression. In the case of 27 - r^3, the degree is 3, indicating that it is a cubic polynomial. Understanding the degree helps in determining the number of roots and the general shape of the polynomial's graph, which is important for further analysis and applications.