Problem 7 - Algebra 2

Question

Simplify each expression. \(\frac{b^{3}-a^{3}}{a^{2}-b^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{b^{2} + ba + a^{2}}{a+b}\).

Step 1: Recognize the Algebraic Identities

In the given expression \(\frac{b^{3}-a^{3}}{a^{2}-b^{2}}\), we can recognize that the numerator \(b^{3}-a^{3}\) is a difference of cubes, and the denominator \(a^{2}-b^{2}\) is a difference of squares. The algebraic identities are:1. Difference of cubes: \(x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})\)2. Difference of squares: \(x^{2} - y^{2} = (x - y)(x + y)\)

Step 2: Apply the Difference of Cubes Formula to the Numerator

Applying the difference of cubes formula to the numerator, we have:\[b^{3} - a^{3} = (b - a)(b^{2} + ba + a^{2})\] Now the expression becomes:\[\frac{(b-a)(b^{2} + ba + a^{2})}{a^{2} - b^{2}}\]

Step 3: Apply the Difference of Squares Formula to the Denominator

Apply the difference of squares formula to the denominator:\[a^{2} - b^{2} = (a - b)(a + b)\]Thus, the expression becomes:\[\frac{(b-a)(b^{2} + ba + a^{2})}{(a-b)(a+b)}\]

Step 4: Simplify the Expression by Canceling Common Terms

Notice that \((b-a) = -(a-b)\), so we can express:\[\frac{(b-a)(b^{2} + ba + a^{2})}{(a-b)(a+b)} = \frac{-(a-b)(b^{2} + ba + a^{2})}{(a-b)(a+b)}\]Now \((a-b)\) is a common term in both the numerator and the denominator and can be canceled out, but remember to keep the negative sign:\[-\frac{b^{2} + ba + a^{2}}{a+b}\]

Step 5: Final Simplified Expression

The expression is now fully simplified. The final result of simplifying the given expression is:\[-\frac{b^{2} + ba + a^{2}}{a+b}\]

Key Concepts

Difference of Cubes

In algebra, the difference of cubes is a special way to factor expressions that have the form \(x^3 - y^3\). This identity helps to simplify expressions by breaking them into a product of simpler expressions. The formula for the difference of cubes is given by:

• \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)

When you encounter an expression like \(b^3 - a^3\), you can treat \(b\) as \(x\) and \(a\) as \(y\). By applying the formula, you rewrite \(b^3 - a^3\) as \((b - a)(b^2 + ba + a^2)\). This reduces the complexity of the original expression, making it easier to solve or simplify further.
The key benefit of using the difference of cubes identity is that it transforms a cumbersome cubic term into a multiplication problem involving simpler terms that can be managed more easily. Understanding how to identify and apply this formula is important because it enables you to handle various polynomial expressions efficiently.

Difference of Squares

The difference of squares is another essential algebraic identity used to simplify expressions. It applies to expressions of the form \(x^2 - y^2\) and follows the formula:

• \(x^2 - y^2 = (x - y)(x + y)\)

For instance, if you have \(a^2 - b^2\) in an expression, you can easily recognize it as a difference of squares. This enables you to express it as the product \((a - b)(a + b)\).
Using this identity is beneficial because it allows for the expression to be broken into factors that are simpler to handle, often leading to terms that can be canceled out or simplified further in a given problem.
Thus, the difference of squares identity is a valuable tool in both simplifying and solving algebraic equations, as it bridges complex quadratic terms to easier linear expressions.

Simplifying Expressions

At the heart of algebra is the process of simplifying expressions, which involves making them as streamlined and manageable as possible. When you are faced with a complex fraction, like the original exercise \(\frac{b^3-a^3}{a^2-b^2}\), your goal is to apply known algebraic identities to simplify the expression.

• Recognize patterns that fit identities (like the difference of cubes and squares).
• Apply those identities to factor the expression.
• Cancel out common terms from the numerator and the denominator.

The importance of simplifying expressions lies in its ability to transform complex algebraic problems into simpler forms, making it easier to solve equations, analyze functions, and understand mathematical relationships.
In the solution provided, recognizing \(b^3-a^3\) as a difference of cubes and \(a^2-b^2\) as a difference of squares allowed these identities to be applied, making the expression ready for simplification. Canceling common terms, when careful attention is paid to sign changes, results in a cleaner, more concise expression.
Mastering this process not only enhances algebraic skills but also prepares students for more advanced mathematical concepts.