Problem 57 - Elementary and Intermediate Algebra

Question

Find the LCD of pair of rational expressions. \(\frac{4 x-5}{x^{2}-4 x-5}, \frac{3 x+1}{x^{2}-25}\)

Step-by-Step Solution

Answer

The LCD is \((x - 5)(x + 1)(x + 5)\).

Step 1: Factor the Denominators

First, factor the denominators of both rational expressions completely.For the first denominator \(x^2 - 4x - 5\), we find factors of \(-5\) that add up to \(-4\), which are \(-5\) and \(1\). The factored form is \((x - 5)(x + 1)\).For the second denominator \(x^2 - 25\), recognize it as a difference of squares: \((x - 5)(x + 5)\).

Step 2: List All Unique Factors

List each unique factor from both denominators. The factors from the first expression are \((x - 5)\) and \((x + 1)\), while the factors from the second expression are \((x - 5)\) and \((x + 5)\). The unique factors are \((x - 5)\), \((x + 1)\), and \((x + 5)\).

Step 3: Construct the LCD

To construct the Lowest Common Denominator (LCD), use each unique factor from Step 2. The LCD must have each unique factor exactly once, as both denominators share the factor \((x - 5)\). Therefore, the LCD is \((x - 5)(x + 1)(x + 5)\).

Key Concepts

Factoring Polynomials

In algebra, factoring polynomials is a crucial skill. It helps break down complex expressions into simpler parts. This process involves finding two or more expressions that multiply together to give the original polynomial. For example:

• Consider the quadratic polynomial, \(x^2 - 4x - 5\).
• We aim to express it as a product of two binomials.
• By seeking numbers that multiply to \(-5\) and add up to \(-4\), we identify \(-5\) and \(1\) as factors.
• The expression factors to \((x - 5)(x + 1)\).

This step simplifies equations, making solving them easier. When you're asked to factor, look for identities, like differences of squares, or use trial and error. Factoring reduces complicated equations into manageable steps.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Similar to numeric fractions, you can perform operations like addition, subtraction, multiplication, and division on them. Simplifying rational expressions often involves:

• Factoring the polynomials in the numerator and the denominator.
• Canceling common factors between them.

For example, in the exercise provided:

• The rational expression is \(\frac{4x - 5}{x^2 - 4x - 5}\).
• By factoring the denominator, we get \(\frac{4x - 5}{(x - 5)(x + 1)}\).

Understanding these steps is key to manipulating and solving polynomial equations effectively. Always factor fully before simplifying; this helps in recognizing cancellation opportunities. Remember, the goal is to work with the simplest form.

Difference of Squares

"Difference of squares" is a special algebraic identity used to factor certain types of polynomials. This identity states that for any numbers \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored into \((a - b)(a + b)\). Here’s how it applies to our example:

• The expression \(x^2 - 25\) is a difference of squares.
• Here, \(a\) is \(x\) and \(b\) is \(5\), thus \(x^2 - 25\) factors to \((x - 5)(x + 5)\).

Recognizing difference of squares is a powerful technique. It allows you to quickly simplify or solve polynomial expressions. Whenever you see a subtraction of two perfect squares, apply this method for a swift factorization. It's a handy tool to have in your algebra toolbox.