Enter The Correct Answer In The Box.Write The Factored Form Of The Least Common Denominator Needed To Simplify This Expression.$\[ \frac{g+1}{g^3+2g-15} + \frac{g+3}{g+5} \\]

Introduction

When dealing with algebraic expressions, it's often necessary to find the least common denominator (LCD) to simplify complex fractions. The LCD is the smallest expression that can be divided by both denominators of the fractions, allowing us to combine them into a single fraction. In this article, we'll explore how to find the LCD of two algebraic expressions and provide a step-by-step guide on how to factor the LCD.

Understanding the Concept of Least Common Denominator (LCD)

The LCD is a fundamental concept in algebra that helps us simplify complex fractions. It's the smallest expression that can be divided by both denominators of the fractions, allowing us to combine them into a single fraction. The LCD is typically denoted by the variable "LCD" or "LCM" (Least Common Multiple).

Step 1: Factor the Denominators

To find the LCD, we need to factor the denominators of both fractions. The first fraction has a denominator of $g^3+2g-15$, which can be factored as $(g-3)(g^2+3g+5)$. The second fraction has a denominator of $g+5$, which is already factored.

Step 2: Identify the Common Factors

Once we have factored the denominators, we need to identify the common factors between the two expressions. In this case, the only common factor is $g+5$.

Step 3: Multiply the Common Factors

To find the LCD, we need to multiply the common factors together. In this case, we multiply $g+5$ by the remaining factors of the first denominator, which are $(g-3)(g^2+3g+5)$.

Step 4: Simplify the LCD

After multiplying the common factors, we need to simplify the LCD by combining like terms. In this case, the LCD is $(g-3)(g+5)(g^2+3g+5)$.

Conclusion

Finding the least common denominator (LCD) is a crucial step in simplifying complex fractions. By following the steps outlined in this article, we can factor the denominators, identify the common factors, multiply the common factors, and simplify the LCD. The LCD is a fundamental concept in algebra that helps us simplify complex fractions and solve equations.

Example Problem

Find the least common denominator (LCD) of the following expression:

{ \frac{g+1}{g^3+2g-15} + \frac{g+3}{g+5} \}

Solution

To find the LCD, we need to factor the denominators of both fractions. The first fraction has a denominator of $g^3+2g-15$, which can be factored as $(g-3)(g^2+3g+5)$. The second fraction has a denominator of $g+5$, which is already factored.

Step 1: Factor the Denominators

The first fraction has a denominator of $g^3+2g-15$, which can be factored as $(g-3)(g^2+3g+5)$. The second fraction has a denominator of $g+5$, which is already factored.

Step 2: Identify the Common Factors

Once we have factored the denominators, we need to identify the common factors between the two expressions. In this case, the only common factor is $g+5$.

Step 3: Multiply the Common Factors

To find the LCD, we need to multiply the common factors together. In this case, we multiply $g+5$ by the remaining factors of the first denominator, which are $(g-3)(g^2+3g+5)$.

Step 4: Simplify the LCD

After multiplying the common factors, we need to simplify the LCD by combining like terms. In this case, the LCD is $(g-3)(g+5)(g^2+3g+5)$.

Final Answer

The final answer is: $\boxed{(g-3)(g+5)(g^2+3g+5)}$

Introduction

Finding the least common denominator (LCD) is a crucial step in simplifying complex fractions. However, it can be a challenging task, especially for students who are new to algebra. In this article, we'll answer some of the most frequently asked questions (FAQs) about finding the LCD.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest expression that can be divided by both denominators of the fractions, allowing us to combine them into a single fraction.

Q: Why is it necessary to find the LCD?

A: Finding the LCD is necessary to simplify complex fractions. By finding the LCD, we can combine the fractions into a single fraction, making it easier to solve equations and simplify expressions.

Q: How do I find the LCD of two fractions?

A: To find the LCD of two fractions, you need to factor the denominators of both fractions. Then, identify the common factors between the two expressions and multiply them together. Finally, simplify the LCD by combining like terms.

Q: What if the denominators of the fractions are not factorable?

A: If the denominators of the fractions are not factorable, you can use the greatest common divisor (GCD) method to find the LCD. The GCD is the largest number that divides both numbers without leaving a remainder.

Q: Can I use a calculator to find the LCD?

A: Yes, you can use a calculator to find the LCD. However, it's always a good idea to check your work by factoring the denominators and multiplying the common factors together.

Q: How do I know if I have found the correct LCD?

A: To check if you have found the correct LCD, you can multiply the LCD by the numerator of each fraction and simplify the expression. If the expression simplifies to a single fraction, then you have found the correct LCD.

Q: Can I use the LCD to solve equations?

A: Yes, you can use the LCD to solve equations. By finding the LCD, you can combine the fractions into a single fraction, making it easier to solve equations.

Q: What are some common mistakes to avoid when finding the LCD?

A: Some common mistakes to avoid when finding the LCD include:

• Not factoring the denominators correctly
• Not identifying the common factors between the two expressions
• Not multiplying the common factors together correctly
• Not simplifying the LCD by combining like terms

Conclusion

Finding the least common denominator (LCD) is a crucial step in simplifying complex fractions. By following the steps outlined in this article and avoiding common mistakes, you can find the correct LCD and simplify complex fractions with ease.

Additional Resources

If you're struggling to find the LCD or need additional help, here are some additional resources that may be helpful:

• Online calculators: There are many online calculators available that can help you find the LCD.
• Algebra textbooks: Algebra textbooks often have examples and exercises that can help you practice finding the LCD.
• Online tutorials: Online tutorials can provide step-by-step instructions and examples to help you learn how to find the LCD.

Final Answer

The final answer is: (g-3)(g+5)(g^2+3g+5)