Problem 74 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{5 a+3 b}{8 a^{2}+2 a b-b^{2}}-\frac{3 a-b}{4 a^{2}-9 a b+2 b^{2}}-\frac{a+5 b}{4 a^{2}+3 a b-b^{2}} $$

Step-by-Step Solution

Answer

Question: Subtract the following rational expressions and simplify the result: $$ \frac{5a+3b}{8a^2 + 2ab - b^2} - \frac{3a-b}{4a^2 - 9ab + 2b^2} - \frac{a+5b}{4a^2 + 3ab - b^2} $$ Answer: $$ \frac{-4a^3 + 18a^2b - 29a^2 - 21ab^2 + 62ab + 15b^3}{(4a + b)(2a - b)(2a + b)(a - b)} $$

Step 1: Find the least common denominator (LCD)

To find the least common denominator, we will factor the denominators of the given expressions: 1. \(8a^2 + 2ab - b^2 = (4a + b)(2a - b)\) 2. \(4a^2 - 9ab + 2b^2 = (2a - b)(2a + b)\) 3. \(4a^2 + 3ab - b^2 = (4a + b)(a - b)\) Now, we can find the least common denominator by multiplying the unique factors: $$LCD = (4a + b)(2a - b)(2a + b)(a - b)$$

Step 2: Rewrite the fractions

Now, rewrite each fraction with the least common denominator: $$ \frac{(5a+3b)(2a+b)(a-b)}{LCD} - \frac{(3a-b)(4a+b)(a-b)}{LCD} - \frac{(a+5b)(2a-b)(2a+b)}{LCD} $$

Step 3: Subtract the numerators

Now, subtract the numerators while keeping the same denominator: $$ \frac{[(5a+3b)(2a+b)(a-b)] - [(3a-b)(4a+b)(a-b)] - [(a+5b)(2a-b)(2a+b)]}{LCD} $$

Step 4: Simplify the numerator

Expanding the numerator and combining like terms: $$ \frac{-4a^3 + 18a^2b - 29a^2 - 21ab^2 + 62ab + 15b^3}{(4a + b)(2a - b)(2a + b)(a - b)} $$ The numerator cannot be factorized further, so this is the final simplified form: $$ \frac{-4a^3 + 18a^2b - 29a^2 - 21ab^2 + 62ab + 15b^3}{(4a + b)(2a - b)(2a + b)(a - b)} $$

Key Concepts

Fractions

Fractions are mathematical expressions that represent the division of one quantity by another. They consist of two parts: the numerator and the denominator. The numerator is the top part, representing the number of parts we have. The denominator is the bottom part, representing the total number of equal parts that make up a whole. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
In algebra, fractions often occur with variables, as seen in rational expressions. Simplifying these fractions involves factoring and canceling common factors in both the numerator and the denominator. This process helps to make the arithmetic operations on fractions easier.
Understanding fractions is crucial in dealing with algebraic operations, especially when combining or subtracting expressions where finding a common denominator is necessary.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. In performing operations like addition or subtraction on fractions, finding a common denominator is essential, as it allows the fractions to be combined. This is particularly important in solving equations involving multiple fractional rational expressions.
To find the least common denominator (LCD), the process typically involves:

• Factoring each denominator into its prime factors.
• Identifying all unique factors across the denominators.
• Multiplying these unique factors together to get the LCD.

In the given problem, denominators are factored, and the LCD is determined by multiplying the unique polynomial factors. Once a common denominator is found, the expressions are rewritten to share this denominator, making subtraction possible.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into a product of simpler polynomials that when multiplied together give the original polynomial. This is analogous to finding the prime factors of a number in arithmetic.
Common methods of factoring polynomials include:

• Factoring by grouping.
• Using the distributive property (e.g., common factor extraction).
• Special formulas such as the difference of squares.

In the context of the exercise, each of the polynomial denominators was factored into two binomials, which made it possible to identify the least common denominator for the algebraic fractions involved. Proper factoring is crucial as it simplifies the process of performing operations between these fractions.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They exhibit behaviors similar to numerical fractions, such as requiring a common denominator for addition or subtraction.
Working with rational expressions involves:

• Factoring the polynomials involved.
• Simplifying the expressions by eliminating common factors.
• Ensuring the domain is defined by excluding values that make the denominator zero.

In the exercise provided, the rational expressions needed to be combined, necessitating the use of a common denominator. Understanding how to manipulate rational expressions is fundamental in algebra, as it applies to various problems including equation solving, integration, and many more advanced math topics.