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Chapter 5: Problem 50

Factor completely. \(a^{2} b^{2}-121\)

Short Answer

Expert verified
(ab + 11)(ab - 11)

Step by step solution

01

Recognize the form of the expression

The given expression is in the form of a difference of squares: a^{2} b^{2} - 121discovers that it matches the patterna^2 - b^2 = (a + b)(a - b).
02

Factor 121

Notice that 121 is a perfect square, as it can be written as a^{2} b^{2} - 11^{2}.
03

Apply the difference of squares formula

Use the difference of squares formula: a^{2} b^{2} - 11^{2} = (ab + 11)(ab - 11).

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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 helpful algebraic concept used to simplify and factor certain types of polynomial expressions. This technique involves recognizing when an expression fits the pattern: \[a^2 - b^2 = (a + b)(a - b)\] This formula tells us that any expression that can be written as the subtraction (or 'difference') of two squared terms can be factored into the product of two binomials.
Perfect square
A perfect square is any number that can be expressed as the product of an integer with itself. In algebra, perfect squares are crucial because they allow us to recognize patterns and factor expressions. For example, 121 is a perfect square since it equals \(11^2\). In our original problem, the expression \(a^2b^2 - 121\) can be rewritten using the perfect square: \(a^2b^2 - 11^2\). Recognizing perfect squares simplifies the process of using algebraic formulas.
Factoring formulas
Factoring formulas are specific algebraic rules that help simplify polynomial expressions. Familiarity with common factoring formulas makes it easier to break down complex expressions. The difference of squares formula is one example. Let's apply it to the given problem:
  1. First, recognize the expression format:

    2. .. Original: \(a^2b^2 - 121\)
    .. Using perfect squares: \(a^2b^2 - 11^2\)
  2. Next, apply the difference of squares formula:

    3. .. \((ab + 11)(ab - 11)\)

This process of recognizing and applying the appropriate factoring formula transforms a complex problem into a straightforward solution. It highlights the importance of mastering factoring techniques in algebra.

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