Question 7Find The LCM Of $6(x-1)^2(x+2)(x-3)$ And $9(x-1)(x+2)^2(x-3)$.A. \$18(x-1)^2(x+2)^2(x-3)$[/tex\]B. $36(x-1)^2(x+2)^2(x-3)^2$C. $3(x-1)(x+2)(x-3)$D.

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Introduction

In mathematics, the least common multiple (LCM) is a concept used to find the smallest multiple that is common to two or more numbers. When dealing with algebraic expressions, finding the LCM can be a bit more complex. In this article, we will explore how to find the LCM of two algebraic expressions, specifically the given expressions $6(x-1)^2(x+2)(x-3)$ and $9(x-1)(x+2)^2(x-3)$.

Understanding the Concept of LCM

Before we dive into finding the LCM of the given expressions, let's briefly review the concept of LCM. The LCM of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 and 6.

Step 1: Factorize the Given Expressions

To find the LCM of the given expressions, we need to factorize them first. The given expressions are:

6(x−1)2(x+2)(x−3)6(x-1)^2(x+2)(x-3)

9(x−1)(x+2)2(x−3)9(x-1)(x+2)^2(x-3)

We can factorize these expressions as follows:

6(x−1)2(x+2)(x−3)=6(x−1)2(x+2)(x−3)6(x-1)^2(x+2)(x-3) = 6(x-1)^2(x+2)(x-3)

9(x−1)(x+2)2(x−3)=32(x−1)(x+2)2(x−3)9(x-1)(x+2)^2(x-3) = 3^2(x-1)(x+2)^2(x-3)

Step 2: Identify the Common Factors

Now that we have factorized the expressions, we can identify the common factors. The common factors are the factors that appear in both expressions. In this case, the common factors are:

(x−1)(x-1)

(x+2)(x+2)

(x−3)(x-3)

Step 3: Determine the LCM

To determine the LCM, we need to take the highest power of each common factor. In this case, the highest power of $(x-1)$ is $2$, the highest power of $(x+2)$ is $2$, and the highest power of $(x-3)$ is $1$.

Therefore, the LCM is:

6(x−1)2(x+2)2(x−3)6(x-1)^2(x+2)^2(x-3)

However, we need to consider the coefficients of the expressions as well. The coefficient of the first expression is $6$, and the coefficient of the second expression is $9$. To find the LCM, we need to take the product of the coefficients and the LCM of the variables.

Therefore, the LCM is:

18(x−1)2(x+2)2(x−3)18(x-1)^2(x+2)^2(x-3)

Conclusion

In conclusion, the LCM of the given expressions $6(x-1)^2(x+2)(x-3)$ and $9(x-1)(x+2)^2(x-3)$ is $18(x-1)2(x+2)2(x-3)$. This is the smallest expression that is a multiple of both given expressions.

Answer

The correct answer is:

Introduction

In our previous article, we explored how to find the least common multiple (LCM) of two algebraic expressions. In this article, we will answer some frequently asked questions (FAQs) related to finding the LCM of algebraic expressions.

Q: What is the least common multiple (LCM)?

A: The LCM is the smallest multiple that is common to two or more numbers. In the context of algebraic expressions, the LCM is the smallest expression that is a multiple of both given expressions.

Q: How do I find the LCM of two algebraic expressions?

A: To find the LCM of two algebraic expressions, you need to follow these steps:

  1. Factorize the given expressions.
  2. Identify the common factors.
  3. Determine the LCM by taking the highest power of each common factor.
  4. Consider the coefficients of the expressions and take the product of the coefficients and the LCM of the variables.

Q: What are the common factors in the LCM?

A: The common factors in the LCM are the factors that appear in both expressions. In the case of the given expressions $6(x-1)^2(x+2)(x-3)$ and $9(x-1)(x+2)^2(x-3)$, the common factors are $(x-1)$, $(x+2)$, and $(x-3)$.

Q: How do I determine the LCM of the variables?

A: To determine the LCM of the variables, you need to take the highest power of each common factor. In the case of the given expressions, the highest power of $(x-1)$ is $2$, the highest power of $(x+2)$ is $2$, and the highest power of $(x-3)$ is $1$.

Q: What is the role of coefficients in finding the LCM?

A: The coefficients of the expressions play a crucial role in finding the LCM. You need to take the product of the coefficients and the LCM of the variables to determine the final LCM.

Q: Can I use the LCM to solve other mathematical problems?

A: Yes, the LCM can be used to solve other mathematical problems, such as finding the greatest common divisor (GCD) or solving systems of equations.

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

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

  • Not factorizing the expressions properly
  • Not identifying the common factors correctly
  • Not taking the highest power of each common factor
  • Not considering the coefficients of the expressions

Conclusion

In conclusion, finding the LCM of algebraic expressions requires careful factorization, identification of common factors, and consideration of coefficients. By following the steps outlined in this article, you can find the LCM of two algebraic expressions and solve other mathematical problems.

Frequently Asked Questions (FAQs)

  • Q: What is the LCM of $2x^2$ and $3x^2$? A: The LCM of $2x^2$ and $3x^2$ is $6x^2$.
  • Q: What is the LCM of $x^2 + 2x$ and $x^2 - 3x$? A: The LCM of $x^2 + 2x$ and $x^2 - 3x$ is $x^2 - x$.
  • Q: What is the LCM of $4x^3$ and $6x^3$? A: The LCM of $4x^3$ and $6x^3$ is $12x^3$.

Answer Key

  • Q: What is the LCM of $2x^2$ and $3x^2$? A: The LCM of $2x^2$ and $3x^2$ is $6x^2$.
  • Q: What is the LCM of $x^2 + 2x$ and $x^2 - 3x$? A: The LCM of $x^2 + 2x$ and $x^2 - 3x$ is $x^2 - x$.
  • Q: What is the LCM of $4x^3$ and $6x^3$? A: The LCM of $4x^3$ and $6x^3$ is $12x^3$.