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: minutes

Problem 73

Textbook Question

Factor each polynomial. See Examples 5 and 6. 8-a^3

Verified step by step guidance

Recognize that the expression \(8 - a^3\) is a difference of cubes.

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

Identify \(x\) and \(y\) in the expression \(8 - a^3\). Here, \(8\) is \(2^3\), so \(x = 2\) and \(y = a\).

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

Simplify the expression: \((2 - a)(4 + 2a + a^2)\).

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 techniques include identifying common factors, using the difference of squares, and applying special formulas like the quadratic formula for second-degree polynomials.

Common Factor

A common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of polynomials, identifying the greatest common factor (GCF) allows for simplification before further factoring. For example, in the polynomial 8 - a^3, the GCF can help in breaking down the expression into simpler multiplicative components.

Difference of Cubes

The difference of cubes is a specific factoring pattern that applies to expressions of the form a^3 - b^3. It can be factored using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). Recognizing this pattern is crucial when dealing with cubic polynomials, as it allows for efficient simplification and solving of equations involving cubic terms.