Problem 55 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{2}\left(x^{2}-1\right)-9\left(x^{2}-1\right) $$

Step-by-Step Solution

Answer

\((x - 1)(x + 1)(x - 3)(x + 3)\)

Step 1: Identify the Common Factor

First, observe that both terms in the expression share a common factor: \(x^2 - 1\). So we can factor \(x^2 - 1\) out of the entire expression. This gives us \((x^2 - 1)(x^2 - 9)\).

Step 2: Factor the First Binomial

Notice that \(x^2 - 1\) is a difference of squares. We can apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Thus, \(x^2 - 1 = (x - 1)(x + 1)\).

Step 3: Factor the Second Binomial

Similarly, \(x^2 - 9\) is also a difference of squares. It can be written as \(x^2 - 3^2\). Applying the difference of squares formula, \(x^2 - 9 = (x - 3)(x + 3)\).

Step 4: Combine All Factors

Now, we can combine all the factors we have identified. The complete factorization of the expression is \((x - 1)(x + 1)(x - 3)(x + 3)\).

Key Concepts

Difference of Squares

In algebra, the concept of "difference of squares" is a straightforward method used to factor certain types of polynomials. It applies when you have an expression in the form of the difference between two perfect squares. For example, the expression \( a^2 - b^2 \) fits this pattern.

To factor a difference of squares, you can use the formula: \[ a^2 - b^2 = (a - b)(a + b) \] This means you take the square root of each squared term, \( a \) and \( b \), and form two binomials: one by subtracting \( b \) from \( a \), and the other by adding \( b \) to \( a \).

In our example, both \( x^2 - 1 \) and \( x^2 - 9 \) are differences of squares. Let's break this down: - In \( x^2 - 1 \), we have \( a = x \) and \( b = 1 \), so it becomes \( (x - 1)(x + 1) \). - In \( x^2 - 9 \), \( a = x \) and \( b = 3 \), resulting in \( (x - 3)(x + 3) \).

Once you are familiar with the pattern of a difference of squares, it becomes a powerful tool for simplifying expressions.

Common Factor

Identifying the "common factor" is an integral step in simplifying algebraic expressions. A common factor is a term that is present in each part of an expression, which can be factored out, making further calculations easier.

For the problem we're working through, observe the expression: \( x^{2}(x^{2}-1)-9(x^{2}-1) \). Both terms, \( x^{2}(x^{2}-1) \) and \( -9(x^{2}-1) \), include the binomial \( x^2 - 1 \). This common factor can be factored out from the entire expression:

• Extracting \( x^2 - 1 \), the expression simplifies to \((x^2 - 1)(x^2 - 9)\).

Once the common factor is factored out, the expression is much simpler and can be further reduced using other factoring techniques, such as the difference of squares.

Binomial Factoring

Binomial factoring involves rewriting a two-term expression (a binomial) as a product of two simpler binomials. This process often uses formulas and identities to simplify expressions.

In our example, once we factor out the common binomial \(x^2 - 1\), we're left with another binomial, \(x^2 - 9\). Both these expressions are prime candidates for binomial factoring since they display the characteristics of a difference of squares.

To factor these binomials, we return to the previously mentioned difference of squares formula:

• The expression \(x^2 - 1\) factors to \((x - 1)(x + 1)\).
• The expression \(x^2 - 9\) factors to \((x - 3)(x + 3)\).

Eventually, the entire expression \((x^2 - 1)(x^2 - 9)\) simplifies to \((x - 1)(x + 1)(x - 3)(x + 3)\).

Binomial factoring, especially when paired with recognizing common factors and differences of squares, greatly streamlines simplifying polynomial expressions.