Select The Correct Answer From Each Drop-down Menu.Consider This Expression:$\[ \frac{9}{x^2-12x} + \frac{7x}{x^2-9} \\]The Least Common Denominator Of The Two Rational Terms Is ( $\square$ )( $\square$ )(

Feb 27, 2025 by ADMIN 210 views

Select the Correct Answer from Each Drop-Down Menu: Finding the Least Common Denominator

Understanding the Problem

When dealing with rational expressions, it's essential to find the least common denominator (LCD) to add or subtract them. The LCD is the smallest expression that both denominators can divide into evenly. In this problem, we're given two rational expressions:

\frac{9}{x^2-12x} + \frac{7x}{x^2-9}

Our goal is to find the LCD of these two expressions.

Factoring the Denominators

To find the LCD, we need to factor the denominators of both expressions. Let's start by factoring the first denominator:

x^2-12x = x(x-12)

Now, let's factor the second denominator:

x^2-9 = (x-3)(x+3)

Finding the Least Common Denominator

The LCD is the product of the highest power of each unique factor that appears in either denominator. In this case, we have:

$x$ from the first denominator
$(x-12)$ from the first denominator
$(x-3)$ from the second denominator
$(x+3)$ from the second denominator

The LCD is the product of these factors:

(x)(x-12)(x-3)(x+3)

Simplifying the LCD

We can simplify the LCD by multiplying the factors together:

(x)(x-12)(x-3)(x+3) = x(x-12)(x^2-9)

Selecting the Correct Answer

Now that we've found the LCD, we can select the correct answer from each drop-down menu:

What is the least common denominator of the two rational terms?
- A) $x(x-12)(x-3)(x+3)$
- B) $x(x-12)(x^2-9)$
- C) $(x-12)(x^2-9)$
- D) $(x-3)(x+3)$
What is the correct answer?
- A) $x(x-12)(x-3)(x+3)$
- B) $x(x-12)(x^2-9)$
- C) $(x-12)(x^2-9)$
- D) $(x-3)(x+3)$

The correct answer is B) $x(x-12)(x^2-9)$ .

Conclusion

Finding the least common denominator is a crucial step when adding or subtracting rational expressions. By factoring the denominators and identifying the unique factors, we can determine the LCD and simplify the expression. In this problem, we found the LCD to be $x(x-12)(x^2-9)$ , which is the correct answer.

Additional Tips and Tricks

When factoring the denominators, make sure to identify the unique factors and their highest powers.
The LCD is the product of the highest power of each unique factor that appears in either denominator.
Simplify the LCD by multiplying the factors together.
When adding or subtracting rational expressions, make sure to use the LCD as the common denominator.

Final Answer

The final answer is B) $x(x-12)(x^2-9)$ .

Understanding the Problem

\frac{9}{x^2-12x} + \frac{7x}{x^2-9}

Our goal is to find the LCD of these two expressions.

Q&A: Finding the Least Common Denominator

Q: What is the least common denominator of the two rational terms?

A: The least common denominator is the smallest expression that both denominators can divide into evenly.

Q: How do I find the least common denominator?

A: To find the least common denominator, you need to factor the denominators of both expressions and identify the unique factors and their highest powers.

Q: What are the unique factors of the denominators?

A: The unique factors of the denominators are:

$x$ from the first denominator
$(x-12)$ from the first denominator
$(x-3)$ from the second denominator
$(x+3)$ from the second denominator

Q: How do I simplify the least common denominator?

A: To simplify the least common denominator, you need to multiply the factors together.

Q: What is the simplified least common denominator?

A: The simplified least common denominator is:

(x)(x-12)(x-3)(x+3) = x(x-12)(x^2-9)

Q: What is the correct answer?

A: The correct answer is B) $x(x-12)(x^2-9)$ .

Common Mistakes to Avoid

Not factoring the denominators correctly
Not identifying the unique factors and their highest powers
Not simplifying the least common denominator correctly

Tips and Tricks

Make sure to factor the denominators correctly
Identify the unique factors and their highest powers
Simplify the least common denominator correctly
Use the least common denominator as the common denominator when adding or subtracting rational expressions

Conclusion

Finding the least common denominator is a crucial step when adding or subtracting rational expressions. By understanding the problem, identifying the unique factors and their highest powers, and simplifying the least common denominator, you can determine the correct answer.

Additional Resources

Final Answer

The final answer is B) $x(x-12)(x^2-9)$ .