Problem 79 - Algebra and Trigonometry

Question

perform the indicated operations. Simplify the result, if possible. $$ \left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}} $$

Step-by-Step Solution

Answer

\[ \frac{(c-d)(-a^{3}+a^{2}+ab+b^{2}+b^{3})}{(a^{3}-b^{3})(a^{2}+ab+b^{2})} \]

Step1: Identify Like Terms and Reorganize

The given expression can be rewritten by reorganizing the terms as follows: \[ \left(\frac{a(c-d) + b(c-d)}{a^{3}-b^{3}} \right) - \frac{c-d}{a^{2}+a b+b^{2}} \]

Step 2: Factor Out Common Factors

The expression can be simplified better by factoring out common factors. So it becomes \[ \frac{(c-d)(a - b)}{a^{3}-b^{3}} - \frac{c-d}{a^{2}+a b+b^{2}} \]

Step 3: Combine Similar Terms

The expression can be combined and simplified into a single fraction as follows: \[ \frac{(a^{2}+ab+b^{2})(c-d)(a - b) - (a^{3}-b^{3})(c-d)}{(a^{3}-b^{3})(a^{2}+ab+b^{2})} \]

Step 4: Simplify Expression

The expression can be simplified better by factoring out common factors. So it becomes \[ \frac{(c-d)(a^{2}+ab+b^{2} - a^{3}+b^{3})}{(a^{3}-b^{3})(a^{2}+ab+b^{2})} \]

Step 5: Further Simplify Expression

After canceling out terms, the simplified expression becomes: \[\frac{(c-d)(-a^{3}+a^{2}+ab+b^{2}+b^{3})}{(a^{3}-b^{3})(a^{2}+ab+b^{2})} \] This is the simplified form of the given expression.

Key Concepts

Factoring Algebraic Expressions

Factoring is essentially the process of breaking down an algebraic expression into the product of its factors. These factors are simpler expressions that, when multiplied together, give you the original expression. For example, factoring the algebraic expression \(a^2 - b^2\) would give you \(a + b) \times (a - b)\).

When factoring expressions like the cubic difference \(a^3 - b^3\), the formula used is \(a - b)(a^2 + ab + b^2)\). This identity helps in simplifying complex fractions. In the context of the exercise, we factored out \(c-d\) and used the difference of cubes to simplify the numerator. By identifying common factors, we create an opportunity to cancel terms later, which leads to a much simpler expression.

Combining Like Terms

Understanding Like Terms

Combining like terms is crucial when simplifying algebraic expressions. Like terms are terms that have the exact same variables raised to the same power. For instance, \(3x^2\) and \(5x^2\) are like terms, while \(3x^2\) and \(5x^3\) are not. Combining like terms involves adding or subtracting coefficients while keeping the variable part unchanged.

During solving the exercise, we identified \(c-d\) as a common factor in multiple terms, which allowed us to combine them. Remember, simplification often involves looking for like terms to combine, as it can significantly reduce the complexity of an expression.

Simplifying Fractions

Reducing Complex Fractions

Simplifying fractions is a fundamental skill in algebra. The goal is to reduce the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. To simplify a fraction, factor both the numerator and the denominator, and then divide out the common factors. This process may involve applying various algebraic identities and formulas.

In the given exercise, after combining like terms and factoring, the complex fraction \(\frac{(c-d)(-a^{3}+a^{2}+ab+b^{2}+b^{3})}{(a^{3}-b^{3})(a^{2}+ab+b^{2})}\) emerged. The simplification hinges on canceling out common factors across the numerator and denominator. The final step of simplifying often reveals a much tidier expression, easier to read and interpret.