Problem 59 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression factors to \((x-1)(x+1)(x-3)(x+3)\).

Step 1: Identify Common Factors

Notice that both terms in the expression \(x^2(x^2 - 1) - 9(x^2 - 1)\) have a common factor of \((x^2 - 1)\). Begin by factoring this expression out of both terms.

Step 2: Factor Out the Common Factor

Factor out the common factor \((x^2 - 1)\) from the expression: \[x^2(x^2 - 1) - 9(x^2 - 1) = (x^2 - 1)(x^2 - 9)\]

Step 3: Recognize the Difference of Squares

Observe that both \(x^2 - 1\) and \(x^2 - 9\) are differences of squares. Recall the formula for difference of squares \(a^2 - b^2 = (a-b)(a+b)\).

Step 4: Factor the Differences of Squares

Factor the expressions using the differences of squares formula:- \(x^2 - 1\) can be written as \((x-1)(x+1)\)- \(x^2 - 9\) can be written as \((x-3)(x+3)\)Thus, the factored form of the expression is:\[(x-1)(x+1)(x-3)(x+3)\]

Step 5: Write the Completely Factored Form

The completely factored form of the original expression is:\[(x-1)(x+1)(x-3)(x+3)\]

Key Concepts

Understanding Common Factors

In algebra, a common factor is a term that appears in each part of an expression and can be factored out, simplifying the expression.
Think of it as finding the "shared pieces" before rearranging the puzzle.
This is the first step to make complex expressions easier to work with. In the expression \(x^{2}(x^{2}-1)-9(x^{2}-1)\), the common factor here is \((x^2 - 1)\). Here's how you spot it:

• Write down each term in the expression: \(x^2(x^2 - 1)\) and \(-9(x^2 - 1)\).
• Notice that \((x^2 - 1)\) appears in both terms.

By factoring \((x^2 - 1)\) out, the expression becomes \((x^2 - 1)(x^2 - 9)\).
This hasn't fully solved our problem yet, but it simplifies the path forward.Next, we'll look at how to break down the second part of our expression using another technique called the "difference of squares."

Exploring Difference of Squares

The difference of squares formula is a powerful algebraic tool which is a way of simplifying expressions like \(a^2 - b^2\). The key idea behind this formula is that whenever you have two square values subtracted, you can rewrite them in factored form.The formula goes like this:

• \(a^2 - b^2 = (a-b)(a+b)\)

Now, let's see it in action with the terms from our expression:

• For \(x^2 - 1\), rewrite it as \(x^2 - 1^2\). This becomes \((x - 1)(x + 1)\).
• For \(x^2 - 9\), rewrite it as \(x^2 - 3^2\). This changes to \((x - 3)(x + 3)\).

By applying this concept, the expression \((x^2 - 1)(x^2 - 9)\) is transformed into \((x-1)(x+1)(x-3)(x+3)\). This step simplifies the factorization process further and resolves our initial algebraic puzzle completely.

Working with Algebraic Expressions

Algebraic expressions can initially seem overwhelming, but you'll find that breaking them down can make solving them much easier.
An algebraic expression is simply a combination of numbers, variables, and operations like addition, subtraction, etc.When factoring expressions like \(x^{2}(x^{2}-1)-9(x^{2}-1)\), keep a few strategies in mind:

• Look for common factors first. They simplify expressions considerably.
• Recognize patterns, such as the difference of squares or perfect square trinomials.
• Rewrite complex expressions into simpler parts to manage them better.

By doing these steps meticulously, you can transform even intimidating expressions into something more manageable, like turning our problem into \((x-1)(x+1)(x-3)(x+3)\).
Algebra is much like a puzzle where every piece (or term) has a place; you just need to find where they fit.