Problem 81 - 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: Understanding the Expression

We need to multiply and simplify the given expression: \((x^2 - a^2)(x^2 + a^2)\). Notice that this resembles the difference of squares.

Step 2: Applying the Difference of Squares Identity

The difference of squares identity is given by \( (p - q)(p + q) = p^2 - q^2 \). In the given expression, treat \( p = x^2 \) and \( q = a^2 \).

Step 3: Substituting into the Identity

Apply the identity: \((x^2 - a^2)(x^2 + a^2) = (x^2)^2 - (a^2)^2\). This results in \(x^4 - a^4\).

Step 4: Result

The expression simplifies to \(x^4 - a^4\). This is the simplest form because there are no further common factors to factor out.

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental algebraic identity that plays a vital role in simplifying and multiplying polynomial expressions. This identity states that for any two terms, say \(p\) and \(q\), the expression \((p - q)(p + q)\) can be simplified to \(p^2 - q^2\).

This identity is very useful when dealing with expressions where the terms are squares. In the case of \((x^2 - a^2)(x^2 + a^2)\), we can identify it as a difference of squares where \(p = x^2\) and \(q = a^2\).

Using the identity:

• The first term in the expanded equation, \(p^2\), becomes \((x^2)^2 = x^4\).
• The second term, \(q^2\), transforms to \((a^2)^2 = a^4\).

The final result following the difference of squares identity is \(x^4 - a^4\). This identity not only simplifies computations but also aids in recognizing hidden patterns in algebraic operations.

Simplifying Expressions

Simplifying expressions involves the process of rewriting them in the most compact, yet equivalent, form. This often means factoring, expanding, or combining like terms.It simplifies calculations and allows for easier comparison of expressions.

For the expression \((x^2 - a^2)(x^2 + a^2)\), recognizing it as a difference of squares helps simplify it.Without simplification, directly multiplying these polynomials could become cumbersome.

The simplified form \(x^4 - a^4\) retains all essential characteristics and relationships of the original expression but in a more concise manner.

• Simplification often reveals the underlying structure of the expression.
• It can expose opportunities for further simplifications.

The process involves looking for patterns or identities, like the difference of squares, that can simplify the task.

Algebraic Identities

Algebraic identities are equations that hold true for any possible values of the variables involved.They provide strategies for simplifying expressions and solving equations.

The difference of squares is just one type of algebraic identity.Identities are like shortcuts that offer insight into polynomial behavior.Important identities include:

• The distributive property: \(a(b + c) = ab + ac\).
• The square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

In our case, \((x^2 - a^2)(x^2 + a^2)\) relies on the identity \(p^2 - q^2\) to simplify into \(x^4 - a^4\).

Using algebraic identities:

• Simplifies tasks by reducing complex expressions to simpler forms.
• Helps verify solutions to polynomial equations.

Profound understanding of these identities can ease your navigation through more advanced algebra problems.