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Chapter 8: Problem 168

Explain how you find the Least Common Denominator of \(x^{2}+5 x+4\) and \(x^{2}-16\).

Short Answer

Expert verified
The LCD of \(x^{2}+5x+4\) and \(x^{2}-16\) is \((x+1)(x+4)(x-4)\).

Step by step solution

01

Factorize the first polynomial

First, factorize the polynomial \(x^{2}+5x+4\). Notice that \(x^{2}+5x+4\) can be factored into \((x+1)(x+4)\).
02

Factorize the second polynomial

Now, factorize the polynomial \(x^{2}-16\). Recognize that \(x^{2}-16\) is a difference of squares, and can be written as \((x-4)(x+4)\).
03

Identify the unique factors

List all the unique factors from both factorizations: \(x+1\), \(x+4\), and \(x-4\).
04

Determine the Least Common Denominator

The Least Common Denominator (LCD) is the product of all unique factors found: \((x+1)(x+4)(x-4)\).

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Key Concepts

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

Polynomial Factorization
To grasp polynomial factorization, it helps to understand the way algebraic expressions are broken down into simpler parts. Factorization involves expressing a polynomial as a product of its factors, which are polynomials of lower degrees. These factors, when multiplied together, give the original polynomial.

For example, in the polynomial given, when we consider the expression \(x^2 + 5x + 4\), we look for two numbers that multiply to 4 (the constant term) and add up to 5 (the coefficient of the middle term). These numbers are 1 and 4, so the polynomial can be written as \((x + 1)(x + 4)\).

Breaking down the process:
  • Identify the constant term and the middle coefficient.
  • Find pairs of numbers that multiply to the constant term.
  • Choose the pair that adds up to the middle coefficient.
  • Rewrite the polynomial using these factors.
Difference of Squares
The difference of squares is a special case in polynomial factorization. It occurs when a polynomial is in the form of \(a^2 - b^2\), which can be factored into \((a - b)(a + b)\).

Let's look at the example polynomial \(x^2 - 16\). Identifying this as a difference of squares involves recognizing that 16 is \(4^2\), making the polynomial \(x^2 - 4^2\).
Using the difference of squares formula, \(a = x\) and \(b = 4\), we factorize the polynomial into \((x - 4)(x + 4)\).

Steps to recognize and factor a difference of squares:
  • Ensure both terms are perfect squares.
  • Express each term as a square of a number or variable.
  • Apply the difference of squares formula to split the expression.
Unique Factors
Unique factors are essential when determining the Least Common Denominator (LCD). After factorizing each polynomial, list all the factors without repetition for those that appear in both expressions.

For the polynomials \(x^2 + 5x + 4\) and \(x^2 - 16\), the factors were identified as follows:
  • For \(x^2 + 5x + 4\) - factors are \((x + 1)(x + 4)\).
  • For \(x^2 - 16\) - factors are \((x - 4)(x + 4)\).

When listing the unique factors, we see the common factor \(x + 4\) is present in both expressions. However, unique factors include \(x+1\), \(x+4\), and \(x-4\). Hence, the LCD for the original polynomials is a product of these unique factors: \((x + 1)(x + 4)(x - 4)\). This ensures that the denominators in the fractions involving these polynomials have a common base.

Summarizing the steps:
  • Factorize each polynomial individually.
  • List all factors from each expression.
  • Identify and list the unique (non-repeating) factors.
  • Use the product of these unique factors to determine the LCD.

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