Problem 39 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(x^{2}-a^{2}\right)\left(x^{2}+a^{2}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^4 - a^4\).

Step 1: Recognize the Expression

The expression given is \((x^2 - a^2)(x^2 + a^2)\). Our goal is to multiply these two expressions and simplify the result.

Step 2: Use Difference of Squares Formula

Recall that the difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). The expression \((x^2 - a^2)\) fits this form, where \(a = x\) and \(b = a\). However, we must first multiply the given expressions directly.

Step 3: Apply FOIL Method

Use the FOIL (First, Outer, Inner, Last) method to multiply the expressions \((x^2 - a^2)(x^2 + a^2)\):1. First: \(x^2 \cdot x^2 = x^4\)2. Outer: \(x^2 \cdot a^2 = x^2a^2\)3. Inner: \(-a^2 \cdot x^2 = -x^2a^2\)4. Last: \(-a^2 \cdot a^2 = -a^4\)

Step 4: Simplify the Expression

Combine the like terms obtained from the FOIL method:- The \(x^2a^2\) and \(-x^2a^2\) terms cancel each other out.- This leaves us with the expression \(x^4 - a^4\).

Step 5: Conclusion

The simplified form of the original expression \((x^2 - a^2)(x^2 + a^2)\) is \(x^4 - a^4\).

Key Concepts

Difference of Squares

The concept of a **difference of squares** is a common pattern in algebra that simplifies the multiplication of two binomials. The general formula is \(a^2 - b^2 = (a-b)(a+b)\). This pattern lets us take an expression that is a difference involving two perfect squares and break it into a simple product. When you see expressions like \((x^2 - a^2)\) - which is our first binomial in the original problem - identify this as a difference of squares. Knowing this property can make initial simplifications quicker once the rest of the multiplication is performed.

FOIL Method

One of the key tools for multiplying two binomials is the **FOIL method**, which stands for First, Outer, Inner, Last. This method ensures that you correctly distribute each term in the first binomial across each term in the second.

• **First:** Multiply the first terms in each binomial together, giving us \(x^2\cdot x^2 = x^4\).
• **Outer:** Multiply the outer terms, \(x^2\times a^2 = x^2a^2\).
• **Inner:** Multiply the inner terms, \(-a^2\times x^2 = -x^2a^2\).
• **Last:** Finally, multiply the last terms together, which yields \(-a^2\times a^2 = -a^4\).

This methodical approach ensures all terms are accounted for and helps prevent mistakes in binomial multiplication.

Simplification

**Simplification** is the process of combining and reducing expressions to their simplest form. After applying the FOIL method, we end up with several terms that need to be simplified. In our example expression \(x^4 + x^2a^2 - x^2a^2 - a^4\), notice that the terms \(x^2a^2\) and \(-x^2a^2\) cancel each other out. This means that they sum to zero and can be removed from the expression. What remains, \(x^4 - a^4\), is the simplified version of the expression. Always combine like terms and look for terms that cancel each other during simplification! It makes your final answer clearer and easier to interpret.

Like Terms

**Like terms** are terms in an expression that have the same variables raised to the same powers. Only like terms can be combined together when simplifying expressions. For instance, in our original problem, the terms \(x^2a^2\) and \(-x^2a^2\) are like because they contain the same variables with identical powers: an \(x^2\) and an \(a^2\). When simplifying expressions, look for terms like these that can cancel each other out or be combined to simplify the expression. This is what allows us to reduce the expression from FOIL to it's simplest form.