Problem 53 - Precalculus Mathematics for Calculus

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 simplified expression is \(x^4 - a^4\).

Step 1: Recognize the Type of Expression

The given expression is of the form \((x^2 - a^2)(x^2 + a^2)\). This problem involves products of two binomials that are similar to a difference of squares and a sum of squares.

Step 2: Apply the Difference of Squares Formula

We use the identity for the difference of squares, which is \((A^2 - B^2) = (A - B)(A + B)\). Here, view \(x^2\) as \(A\) and \(a^2\) as \(B\). However, the expression involves both a sum and a difference, so we will expand instead of directly applying this.

Step 3: Expand the Expression

Multiply the binomials together: \((x^2 - a^2)(x^2 + a^2)\). Use the distributive property. Specifically, this looks like:\((x^2)(x^2 + a^2) + (-a^2)(x^2 + a^2)\).

Step 4: Distribute Each Term

First, distribute \(x^2\) across \((x^2 + a^2)\), which gives:\[x^2 \cdot x^2 + x^2 \cdot a^2 = x^4 + x^2a^2\]Next, distribute \(-a^2\) across \((x^2 + a^2)\), which gives:\[-a^2 \cdot x^2 - a^2 \cdot a^2 = -a^2x^2 - a^4\]

Step 5: Combine Like Terms

Combine all the terms obtained:\[x^4 + x^2a^2 - a^2x^2 - a^4\]The middle terms \(x^2a^2\) and \(-a^2x^2\) are like terms and cancel each other out. This simplifies to:\[x^4 - a^4\]

Step 6: Identify and Simplify Final Result

Recognize that the final simplified form, \(x^4 - a^4\), is a difference of squares: \((x^2)^2 - (a^2)^2\), which is already the simplest form for this problem.

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra. It involves expressions that can be written in the form \(A^2 - B^2\). This type of expression can be factored using the identity:

• \(A^2 - B^2 = (A + B)(A - B)\)

This is useful for simplifying polynomial expressions and solving equations. In the given problem, we started with an expression that resembles the difference of squares.By recognizing this pattern, it guides us in deciding how to manipulate and simplify the expression. Although our expression also included a sum of squares \((x^2 + a^2)\), the difference of squares concept still plays an important role.Using it helps us understand the structure of our problem, even if we need to expand and combine the terms later.

Binomial Multiplication

Binomial multiplication is another key idea here. A binomial is a polynomial with two terms, such as \((x - y)\) or \((m + n)\). When multiplying two binomials, the distributive property is applied.Consider the expression

• \((x^2 - a^2)(x^2 + a^2)\)

To multiply these binomials:

• Distribute the first term of the first binomial \((x^2)\) over the second binomial \((x^2 + a^2)\)
• Then do the same with the second term \((-a^2)\)

This process is akin to using the FOIL method (First, Outside, Inside, Last) when multiplying binomials. It ensures all terms are accounted for.The expanded form can then be rearranged and simplified by combining like terms.

Simplifying Expressions

Simplifying expressions is the final step in many algebra problems. It involves combining and reducing terms to form the simplest version of the expression.When simplifying, focus on:

• Combining like terms (terms that have the same variable raised to the same power)
• Cancelling terms where possible, such as \(x^2a^2 - a^2x^2\) which results in zero
• Recognizing patterns, such as the difference of squares, to rewrite expressions in their simplest form

In our problem, we ended with \(x^4 - a^4\). Recognizing it as a difference of squares, we see it as \((x^2)^2 - (a^2)^2\).Although \(x^4 - a^4\) is already its simplest form, understanding these simplification steps is crucial.They streamline complex expressions and make further algebraic operations easier.