Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

Previous problemNext problem

2:06 minutes

Problem 74

Textbook Question

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

Verified step by step guidance

Recognize that the expression \(27 - r^3\) is a difference of cubes.

Recall the formula for factoring a difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Identify \(a\) and \(b\) in the expression \(27 - r^3\). Here, \(a = 3\) and \(b = r\) because \(27 = 3^3\) and \(r^3 = r^3\).

Apply the difference of cubes formula: \((3 - r)(3^2 + 3r + r^2)\).

Simplify the expression: \((3 - r)(9 + 3r + r^2)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.