Problem 48 - Beginning and Intermediate Algebra

Question

Factor completely. $$121-w^{2}$$

Step-by-Step Solution

Answer

The short answer is \(121-w^2 = (11-w)(11+w)\).

Step 1: Identify the squares

First, let's identify the squares in the expression. In the given expression \(121-w^2\), 121 is a perfect square, with square root equal to 11, and \(w^2\) is also a square, with square root equal to \(w\).

Step 2: Apply the difference of squares formula

Now that we've identified the squares, we can apply the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\). In our case, \(a = 11\) and \(b = w\), so we can replace those values into the formula:

Step 3: Substitute the values of a and b into the formula

Substitute \(a = 11\) and \(b = w\) into the difference of squares formula: \((11-w)(11+w)\)

Step 4: Write the complete factorization

The complete factorization of the given expression is: \(121-w^2 = (11-w)(11+w)\) This is our final factored form of the given expression.

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down complex expressions into simpler, more manageable pieces. Think of it as finding pieces that multiply together to give you the original expression. This process is crucial because it helps simplify complex problems, making them easier to solve or work with further.
Here’s a general approach to factoring:

• Look for common factors between terms.
• Identify any special formulas, like difference of squares or perfect square trinomials.
• Break down the expression using these methods.

In the exercise we looked at earlier, the expression was factored using the difference of squares formula, which is a special case of factoring.

Perfect Squares

Perfect squares are numbers or expressions that can be written as the product of an integer or a variable with itself. They are especially helpful in algebra because they simplify into neat results.
For example, the number 121 is a perfect square because it is equal to 11 times 11. Similarly, a variable like \(w^2\) is a perfect square because it can be written as \(w\times w\).
Recognizing perfect squares is the first step in applying certain factoring techniques. In problems involving difference of squares, identifying these perfect squares allows you to rewrite and factor expressions simply by knowing their square roots. Understanding this concept is key to smoothly navigating algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are the building blocks of algebra and appear in many different forms.
The process of manipulating these expressions involves performing operations such as addition, subtraction, multiplication, division, and factoring.

• Types: They can be as simple as a single number or variable, or as complex as a combination of different terms and operations.
• Simplifying: One of the main objectives when working with algebraic expressions is to simplify them, which may involve factoring or using algebraic identities.

Understanding how to interpret and manipulate algebraic expressions is a fundamental skill in mathematics, allowing for better problem-solving and comprehension of more complex algebraic concepts.