Problem 56 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$\left(a^{2}-1\right) b^{2}-4\left(a^{2}-1\right)$$

Step-by-Step Solution

Answer

\((a - 1)(a + 1)(b - 2)(b + 2)\)

Step 1: Identify the Common Factor

First, observe the given expression \( (a^2 - 1)b^2 - 4(a^2 - 1) \). Notice that \( (a^2 - 1) \) is a common factor in both terms of the expression. Factoring \( (a^2 - 1) \) out of the expression, we obtain: \((a^2 - 1)(b^2 - 4)\).

Step 2: Factor the Difference of Squares

Next, look at the new expression \((a^2 - 1)(b^2 - 4)\). Both \(a^2 - 1\) and \(b^2 - 4\) are difference of squares which can be factored further. The factorization for \(a^2 - 1\) is \((a - 1)(a + 1)\) and for \(b^2 - 4\) is \((b - 2)(b + 2)\).

Step 3: Write the Fully Factored Expression

Assemble the completely factored form by substituting the factored forms from Step 2 into the expression. The fully factored expression is: \((a - 1)(a + 1)(b - 2)(b + 2)\).

Key Concepts

Factoring expressions

Factoring expressions is an essential technique in algebra that simplifies complicated algebraic expressions. Essentially, it involves rewriting a sum or difference of terms into a product of simpler components. This makes it easier to solve equations or simplify expressions. Let's break down this concept a little further.

• It involves finding the highest common factors of the terms in an expression and expressing them as a product.
• Factoring allows mathematicians and students to simplify equations or make apparent structures in algebraic expressions that may not be immediately visible.

Finding the factors means identifying values or expressions that can multiply to recreate the original expression. This becomes important in solving quadratic equations or simplifying polynomial expressions.

Difference of squares

The difference of squares is a specific pattern in algebra where two squared terms are subtracted from each other. It follows the identity:\[a^2 - b^2 = (a - b)(a + b)\].

This pattern is significant because it allows immediate factorization, simplifying the process. Identifying a difference of squares often means recognizing two perfect squares separated by a subtraction sign. Here's why it's useful:

• It reduces complex expressions into simple, more manageable factors.
• Allows quicker solutions to algebraic problems that fit this pattern.

In practice, whenever you spot terms like \(a^2 - 1\) or \(b^2 - 4\), these are prime candidates for the difference of squares, leading to quick and effective factorization.

Common factor

A common factor is a term that divides all terms of an expression without leaving a remainder, and it can be factored out to simplify expressions significantly. Identifying common factors is crucial in the factoring process, as it lays the groundwork for more complex operations, such as factoring the difference of squares.

The steps to identify and use common factors include:

• Scanning the expression to see if there’s a term that appears in all components.
• Dividing each term by this common factor, effectively simplifying the entire expression.

In the step-by-step solution provided above, the expression \((a^2 - 1)\) was discovered as a common factor in both terms of the given problem. Factoring out enables further simplification and helps display the underlying structure of the algebraic expression.