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Chapter 8: Problem 172

In the following exercises, find the LCD. $$ \frac{6}{a^{2}+14 a+45}, \frac{5 a}{a^{2}-81} $$

Short Answer

Expert verified
The LCD is \( (a + 9)(a + 5)(a - 9) \).

Step by step solution

01

- Factor the first denominator

The first denominator is \( a^2 + 14a + 45 \). To factor this quadratic expression, look for two numbers that multiply to 45 and add to 14. These numbers are 9 and 5. Thus, \( a^2 + 14a + 45 = (a + 9)(a + 5) \).
02

- Factor the second denominator

The second denominator is \( a^2 - 81 \). This is a difference of squares, which can be factored as \( (a + 9)(a - 9) \).
03

- Identify the Least Common Denominator

The factors of the first denominator are \( (a + 9)(a + 5) \) and the factors of the second denominator are \( (a + 9)(a - 9) \). The LCD must include all unique factors from both denominators: \( (a + 9)(a + 5)(a - 9) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring quadratics
Quadratic expressions are polynomials of the form \(ax^2 + bx + c\). Factoring quadratics involves transforming them into the product of two binomials. Here's how to do it:
  • Identify a, b, and c in the expression.
  • Find two numbers that multiply to \(a\cdot c\) and add up to b.
  • Rewrite the quadratic expression using these two numbers to split the middle term.
  • Factor by grouping.
In the given exercise, the expression \(a^2 + 14a + 45\) is factored by looking for two numbers that multiply to 45 and add to 14. These numbers are 9 and 5. Therefore, the factorization is: \(a^2 + 14a + 45 = (a + 9)(a + 5)\). Simple practice can make you adept at spotting these pairs quickly!
difference of squares
A special type of quadratic is the difference of squares, which is in the form \(a^2 - b^2\). This can always be factored into \((a + b)(a - b)\). This method is concise and only requires recognizing the form.
  • Identify if the expression is in the difference of squares form.
  • Express it as two binomials, one addition and one subtraction.
In the exercise, the second denominator \(a^2 - 81\) fits this pattern, as 81 is \(9^2\). Therefore, it factors to \((a + 9)(a - 9)\). By knowing this method, you can simplify many problems quicker.
rational expressions
Rational expressions are fractions where the numerator and denominator are polynomials. To work with them efficiently, it's important to factor the polynomials and find a common denominator.
The LCD (Least Common Denominator) is the smallest expression that all denominators can divide into evenly. Finding the LCD involves:
  • Factoring each denominator completely.
  • Identifying all unique factors from all denominators.
  • Combining these factors into a single expression.
In this exercise, factoring gives us \((a + 9)(a + 5)\) and \((a + 9)(a - 9)\). The LCD must include every unique factor: \((a + 9)(a + 5)(a - 9)\). This ensures that both original denominators can divide into the LCD without leaving a remainder.
Understanding how to find the least common denominator helps to combine and simplify rational expressions.

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