Problem 4 - Elementary and Intermediate Algebra

Question

The factorizations of the denominators of two rational expressions are given. Find the LCD. a. \(12 a=2 \cdot 2 \cdot 3 \cdot a\) \(18 a^{2}=2 \cdot 3 \cdot 3 \cdot a \cdot a\) b. \(x^{2}-36=(x+6)(x-6)\) \(3 x-18=3(x-6)\)

Step-by-Step Solution

Answer

a. LCD = 36a² b. LCD = 3(x+6)(x-6)

Step 1: Identify the Unique Factors for Part (a)

For the expression \(12a\), the factorization is \(2 \cdot 2 \cdot 3 \cdot a\), including factors \{2, 3, a\}. For the expression \(18a^2\), the factorization is \(2 \cdot 3 \cdot 3 \cdot a \cdot a\), with factors \{2, 3, a\}. Collect all unique factors from both expressions: \{2, 3, a\}.

Step 2: Determine the Maximum Power for Part (a)

Examine the powers of each factor in both expressions. For \(2\), both have a power of \(2^1\). For \(3\), \(18a^2\) has \(3^2\) compared to \(3^1\) in \(12a\). For \(a\), \(18a^2\) has \(a^2\) while \(12a\) has \(a^1\). The LCD is \(2^2 \cdot 3^2 \cdot a^2\), which gives \(36a^2\).

Step 3: Identify the Unique Factors for Part (b)

The factorization of \(x^2-36\) is \((x+6)(x-6)\), having factors \{x+6, x-6\}. The factorization of \(3x-18\) is \(3(x-6)\), including factors \{3, x-6\}. Collect the unique factors: \{3, x+6, x-6\}.

Step 4: Determine the Maximum Power for Part (b)

All factors are present as single powers, no repeated factors with greater powers. Thus, the LCD is the product of all unique factors: \(3(x+6)(x-6)\).

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like fractions involving numbers, rational expressions can be simplified, added, subtracted, multiplied, or divided.
When dealing with rational expressions, particularly when adding or subtracting them, finding a common denominator is necessary. This allows us to combine the expressions into a single fraction.

• Numerators and denominators are polynomials.
• Operations work similarly to arithmetic fractions.
• Finding a least common denominator (LCD) simplifies addition and subtraction.

Understanding the factors of these expressions helps in determining the LCD, which ensures more straightforward operations. By focusing on the polynomial denominators, you can better manage these rational expressions.

Factorization

Factorization is the process of breaking down a complex expression into a product of simpler factors. This can help simplify expressions and find common denominators in rational expressions.
For instance, the factorization of a number like 12 is breaking it down to its prime factors: \(2 \cdot 2 \cdot 3\). In algebra, we apply similar principles to polynomials. For example, a quadratic equation \(x^2 - 36\) can be factored as \((x+6)(x-6)\).

• Prime factorization breaks numbers into primes.
• Polynomial factorization applies similar rules to algebraic expressions.
• Used to simplify expressions and find LCDs in rational expressions.

By identifying these basic building blocks within an expression, one can simplify and manipulate these expressions more effectively, making complex problems more manageable.

Polynomial Fractions

Polynomial fractions are expressions where both the numerator and the denominator are polynomials. These can pose challenges similar to those faced with regular fractions, such as finding common denominators for addition or subtraction.
The process involves simplifying or finding the least common denominator (LCD) by factorization. Suppose we have two fractions with polynomial denominators, factorizing each helps to find their common ground. A critical skill in managing polynomial fractions is distinct identification and combination of factors.

• Both top and bottom are polynomials.
• Operations require finding common denominators.
• Factorization aids in simplification and combination.

Finding the LCD involves gathering all unique factors of the polynomial denominators and then ensuring each factor is raised to the highest power present in any of the denominators. This step is crucial in successfully simplifying and solving polynomial fraction problems.