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Chapter 7: Problem 148

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

Short Answer

Expert verified
The LCD is \( (x + 1)(x + 4)(x - 4) \).

Step by step solution

01

- Factor the first polynomial

To begin, factor the polynomial \(x^{2} + 5x + 4\). The factors of the polynomial are found by looking for two numbers that multiply to 4 (the constant term) and add to 5 (the coefficient of the linear term). These numbers are 1 and 4, so the factors are \( (x + 1)(x + 4) \).
02

- Factor the second polynomial

Next, factor the polynomial \(x^{2} - 16\). This is a difference of squares, which factors into \((x + 4)(x - 4)\).
03

- Identify the unique factors

The factors of \(x^{2} + 5x + 4\) are \( (x + 1)(x + 4) \), and the factors of \(x^{2} - 16\) are \( (x + 4)(x - 4) \). The unique factors from both polynomials are \( (x + 1), (x + 4), \text{and} (x - 4) \).
04

- Multiply the unique factors

To find the Least Common Denominator (LCD), multiply the unique factors together: \( (x + 1)(x + 4)(x - 4) \).

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Key Concepts

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

factoring polynomials
Factoring polynomials is a process where you break down a polynomial into simpler expressions, called 'factors,' that when multiplied together give the original polynomial. This process often makes solving equations easier.

For instance, to factor the polynomial \(x^2 + 5x + 4\), we look for two numbers that multiply to 4 (the constant term) and add to 5 (the coefficient of the middle term). These numbers are 1 and 4. So, \(x^2 + 5x + 4\) factors into \((x + 1)(x + 4)\).

Remember, the key steps in factoring are:
  • Identify the constant term and the coefficient of the middle term.
  • Find two numbers that multiply to the constant and add to the middle term's coefficient.
  • Re-write the polynomial in its factored form.
Practice these steps to become proficient at factoring polynomials.
difference of squares
The difference of squares is a special case in polynomial factoring. It occurs when you have an expression in the form \(a^2 - b^2\). This form can always be factored into \((a + b)(a - b)\).

In our example, the polynomial \(x^2 - 16\) is a difference of squares because it can be rewritten as \((x^2) - (4^2)\). Using the difference of squares formula, it factors into \((x + 4)(x - 4)\).

To factor a difference of squares, always:
  • Check if the polynomial is of the form \(a^2 - b^2\).
  • Rewrite the polynomial showing the squared terms.
  • Apply the formula \((a + b)(a - b)\).
Mastering this method allows you to tackle a wide range of polynomial factoring problems.
unique factors
When working with polynomials, especially in finding the Least Common Denominator (LCD), identifying unique factors is crucial. The unique factors are the distinct elements that, when multiplied, reconstruct the original polynomials.

In our case, after factoring \(x^2 + 5x + 4\) into \((x + 1)(x + 4)\) and \(x^2 - 16\) into \((x + 4)(x - 4)\), we compare the sets of factors. Here, the unique factors are \((x + 1), (x + 4),\) and \((x - 4)\).

To find the LCD, follow these steps:
  • Factorize each polynomial completely.
  • List all the factors of each polynomial.
  • Identify the unique factors from the list.
  • Multiply these unique factors together.
This will give you the LCD, which encapsulates all necessary factors from the polynomials involved.

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Most popular questions from this chapter

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Solve. $$\frac{R}{T}=W \text { for } T$$

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