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

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1:58 minutes

Problem 61

Textbook Question

Factor each polynomial. See Examples 5 and 6. x^4-16

Verified step by step guidance

Recognize that the expression \(x^4 - 16\) is a difference of squares.

Recall the formula for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).

Identify \(a^2 = x^4\) and \(b^2 = 16\), so \(a = x^2\) and \(b = 4\).

Apply the difference of squares formula: \(x^4 - 16 = (x^2 - 4)(x^2 + 4)\).

Notice that \(x^2 - 4\) is also a difference of squares, which can be further factored as \((x - 2)(x + 2)\).

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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 simpler polynomials or factors. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Common techniques include identifying common factors, using special product formulas, and applying methods like grouping or synthetic division.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions of the form a^2 - b^2, which can be factored into (a - b)(a + b). In the case of the polynomial x^4 - 16, it can be recognized as a difference of squares since 16 is 4^2, allowing for further factoring into simpler components.

Quadratic Form

A polynomial can sometimes be expressed in a quadratic form, which is a polynomial of degree two. In the case of x^4 - 16, after applying the difference of squares, it can be factored into (x^2 - 4)(x^2 + 4). The term x^2 - 4 can be further factored as another difference of squares, illustrating the importance of recognizing quadratic structures in higher-degree polynomials.