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Chapter 0: Problem 22

$$ \text { Factor the polynomials in Exercises } \text { . } $$ $$ 4 x^{2}-1 $$

Short Answer

Expert verified
(2x - 1)(2x + 1)

Step by step solution

01

Recognize the Form

Identify the given polynomial as a difference of squares. The general form for a difference of squares is \[ a^2 - b^2 = (a-b)(a+b) \]
02

Identify 'a' and 'b'

Rewrite each term of the polynomial as a square: \[ 4x^2 - 1 = (2x)^2 - 1^2\] Here, \(a = 2x\) and \(b = 1\)
03

Apply the Difference of Squares Formula

Use the difference of squares formula to factor the polynomial: \[ (a - b)(a + b) = (2x - 1)(2x + 1) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a special polynomial form that makes factoring easy. It means subtracting one squared number from another: a^2 - b^2. This can be factored into two binomials: (a - b)(a + b). Recognizing this form is crucial for quick factoring.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In our problem, the symbols are the terms in the polynomial: 4x^2 and 1. The manipulation involves rewriting these terms to make the factoring process simpler. For instance, rewriting 4x^2 as (2x)^2 allows us to recognize it as part of a difference of squares.
Polynomial Identities
Polynomial identities are equations that are true for all values of the variables. They help in simplifying and solving polynomial expressions. The difference of squares identity: a^2 - b^2 = (a - b)(a + b) is a common tool used in factoring polynomials. By recognizing and applying this identity, we can factor complicated expressions quickly and easily.
Factoring Techniques
There are various factoring techniques, and choosing the right one can make problems easier to solve. The difference of squares is one such technique. Here's a quick step-by-step technique to factor:
  • Identify the polynomial form.
  • Rewrite each term as a square if possible.
  • Apply the appropriate factoring identity.
For example, 4x^2 - 1 is identified as a difference of squares and rewritten as (2x)^2 - 1^2. Using the formula, we directly factor it into (2x - 1)(2x + 1).

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Most popular questions from this chapter

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