Select The Correct Answer.What Is The Quotient Of $\frac{3x^2-12x+30}{3x}$?A. $\frac{x^2-4x+10}{x}$ B. $\frac{x^2-4x+10}{3}$ C. $x^2-4x+10$ D. $\frac{3(x^2-4x+10)}{x}$
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students to master. In this article, we will focus on simplifying a specific type of algebraic expression, namely the quotient of a polynomial divided by another polynomial. We will use the given problem as a case study to demonstrate the step-by-step process of simplifying algebraic expressions.
The Problem
The problem asks us to find the quotient of the expression . To solve this problem, we need to simplify the given expression by dividing the numerator by the denominator.
Step 1: Factor the Numerator
The first step in simplifying the expression is to factor the numerator. We can start by factoring out the greatest common factor (GCF) of the terms in the numerator.
3x^2 - 12x + 30 = 3(x^2 - 4x + 10)
Step 2: Divide the Numerator by the Denominator
Now that we have factored the numerator, we can divide it by the denominator. To do this, we need to divide the factored form of the numerator by the denominator.
\frac{3(x^2 - 4x + 10)}{3x} = \frac{3}{3} \cdot \frac{(x^2 - 4x + 10)}{x}
Step 3: Simplify the Expression
Now that we have divided the numerator by the denominator, we can simplify the expression further. We can cancel out the common factor of 3 in the numerator and denominator.
\frac{3}{3} \cdot \frac{(x^2 - 4x + 10)}{x} = 1 \cdot \frac{(x^2 - 4x + 10)}{x}
Step 4: Write the Final Answer
The final answer is the simplified form of the expression.
\frac{(x^2 - 4x + 10)}{x} = \frac{x^2-4x+10}{x}
Conclusion
In this article, we have demonstrated the step-by-step process of simplifying algebraic expressions. We have used the given problem as a case study to show how to factor the numerator, divide it by the denominator, and simplify the expression. The final answer is .
Answer Options
The answer options are:
A. B. C. D.
Which Answer is Correct?
The correct answer is A. .
Why is this the Correct Answer?
This is the correct answer because we have simplified the expression by factoring the numerator, dividing it by the denominator, and canceling out the common factor of 3. The final answer is the simplified form of the expression.
What is the Importance of Simplifying Algebraic Expressions?
Simplifying algebraic expressions is an essential skill for students to master because it helps them to:
- Understand the structure of algebraic expressions
- Identify the key components of an expression
- Simplify complex expressions
- Solve problems more efficiently
Conclusion
Introduction
In our previous article, we demonstrated the step-by-step process of simplifying algebraic expressions. In this article, we will provide a Q&A guide to help students understand the concepts and techniques involved in simplifying algebraic expressions.
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to factor the numerator. This involves identifying the greatest common factor (GCF) of the terms in the numerator and factoring it out.
Q: How do I factor the numerator?
A: To factor the numerator, you need to identify the GCF of the terms in the numerator. You can do this by looking for the largest factor that divides all the terms in the numerator. Once you have identified the GCF, you can factor it out of the numerator.
Q: What is the next step in simplifying an algebraic expression?
A: The next step in simplifying an algebraic expression is to divide the numerator by the denominator. This involves dividing the factored form of the numerator by the denominator.
Q: How do I divide the numerator by the denominator?
A: To divide the numerator by the denominator, you need to divide the factored form of the numerator by the denominator. You can do this by canceling out any common factors between the numerator and denominator.
Q: What is the final step in simplifying an algebraic expression?
A: The final step in simplifying an algebraic expression is to write the simplified form of the expression. This involves combining any remaining terms in the numerator and denominator.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not factoring the numerator correctly
- Not canceling out common factors between the numerator and denominator
- Not combining remaining terms in the numerator and denominator
Q: How can I practice simplifying algebraic expressions?
A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.
Q: What are some real-world applications of simplifying algebraic expressions?
A: Simplifying algebraic expressions has many real-world applications, including:
- Solving problems in physics and engineering
- Modeling population growth and decay
- Analyzing data and making predictions
Q: Why is it important to simplify algebraic expressions?
A: Simplifying algebraic expressions is important because it helps you to:
- Understand the structure of algebraic expressions
- Identify the key components of an expression
- Simplify complex expressions
- Solve problems more efficiently
Conclusion
In conclusion, simplifying algebraic expressions is an essential skill for students to master. By following the step-by-step process outlined in this article, students can simplify complex expressions and solve problems more efficiently. We hope this Q&A guide has been helpful in answering your questions and providing you with a better understanding of the concepts and techniques involved in simplifying algebraic expressions.
Frequently Asked Questions
- Q: What is the difference between simplifying and factoring an algebraic expression? A: Simplifying an algebraic expression involves combining like terms and canceling out common factors, while factoring an algebraic expression involves expressing it as a product of simpler expressions.
- Q: How do I know when to simplify an algebraic expression? A: You should simplify an algebraic expression when it is necessary to make the expression more manageable or to solve a problem more efficiently.
- Q: Can I simplify an algebraic expression that has a variable in the denominator? A: Yes, you can simplify an algebraic expression that has a variable in the denominator, but you need to be careful to avoid dividing by zero.
Additional Resources
- Algebraic Expression Simplification Worksheet: A worksheet with examples and exercises to help you practice simplifying algebraic expressions.
- Algebraic Expression Simplification Video: A video tutorial that demonstrates the step-by-step process of simplifying algebraic expressions.
- Algebraic Expression Simplification Online Practice Test: An online practice test that allows you to practice simplifying algebraic expressions and track your progress.