Problem 46 - Beginning and Intermediate Algebra

Question

Factor completely. $$4-b^{2}$$

Step-by-Step Solution

Answer

The expression \(4-b^{2}\) can be factored completely as \((2 - b)(2 + b)\).

Step 1: Identify a and b

From the given expression, we can see that: \(a^2 = 4\) and \(b^2 = b^2\) Now we can find the values of a and b. Since \(a^2 = 4\), taking the square root of both sides, we get \(a = 2\). For \(b^2 = b^2\), we already have the value of b.

Step 2: Apply the difference of squares formula

Using the formula \(a^2 - b^2 = (a - b)(a + b)\), we can substitute the values of a and b: \((2 - b)(2 + b)\) So, the factored expression is \((2 - b)(2 + b)\).

Key Concepts

Difference of Squares

The difference of squares is a special type of polynomial expression that can be written in the form \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares. This configuration is unique because it can be easily factored into two binomials: \((a - b)(a + b)\).
To apply the difference of squares:

• Identify if the expression is in the form of \(a^2 - b^2\).
• Recognize that \(a\) and \(b\) are the square roots of the two terms.
• Factor the expression using the formula \((a - b)(a + b)\).

The simplicity of this method makes it a powerful tool in algebra. By recognizing the pattern, it is possible to quickly and correctly factor polynomials that fit the difference of squares scenario.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables, like \(x^2 + 5x + 6\). Each part of a polynomial, such as \(x^2\) or \(5x\), is known as a 'term'. Polynomials can be classified based on the number of terms they have:

• Monomial: A single term, such as \(3x\).
• Binomial: Two terms, like \(x^2 - 4\).
• Trinomial: Three terms, example \(x^2 + 5x + 6\).

Polynomials are used extensively across numerous areas of mathematics due to their versatility and capability to represent a wide array of functions and relationships.
Recognizing and working with polynomial expressions is essential because it aids in simplifying equations and functions, solving for unknowns, and understanding mathematical relationships.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions and equations. This technique is critical for solving equations and for transforming expressions into more useful or interpretable forms.
Some key aspects of algebraic manipulation include:

• Identifying like terms to simplify expressions.
• Applying mathematical operations correctly, such as addition, subtraction, multiplication, and division.
• Utilizing formulas, like the difference of squares, to factor expressions.

In our context of factoring \(4 - b^2\), algebraic manipulation is used by applying the difference of squares pattern. Once the terms \(4\) and \(b^2\) are recognized as squares, they are factored into the product of two binomials: \((2 - b)(2 + b)\). The goal of algebraic manipulation is to make expressions more manageable and solvable, whether they involve simple or complex algebraic procedures.