Enter The Correct Answer In The Box.What Is The Simplest Form Of The Expression Representing This Product? X + 10 X 2 + 7 X − 18 ⋅ 3 X 2 − 12 X + 12 3 X + 30 \frac{x+10}{x^2+7x-18} \cdot \frac{3x^2-12x+12}{3x+30} X 2 + 7 X − 18 X + 10 ​ ⋅ 3 X + 30 3 X 2 − 12 X + 12 ​ □ \square □

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Introduction

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Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given problem: $\frac{x+10}{x^2+7x-18} \cdot \frac{3x^2-12x+12}{3x+30}$. We will break down the solution into manageable steps, using clear explanations and examples to illustrate each concept.

Understanding the Problem

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The given problem involves multiplying two algebraic expressions, which can be simplified by factoring and canceling common terms. To begin, let's examine the expressions individually:

• $\frac{x+10}{x^2+7x-18}$
• $\frac{3x^2-12x+12}{3x+30}$

Factoring the Numerators and Denominators

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To simplify the expressions, we need to factor the numerators and denominators. Let's start with the first expression:

• $\frac{x+10}{x^2+7x-18}$

We can factor the numerator as $(x+10)$ and the denominator as $(x+9)(x-2)$.

(x+10) / ((x+9)(x-2))

Similarly, for the second expression:

• $\frac{3x^2-12x+12}{3x+30}$

We can factor the numerator as $3(x^2-4x+4)$ and the denominator as $3(x+10)$.

3(x^2-4x+4) / (3(x+10))

Canceling Common Terms

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Now that we have factored the expressions, we can cancel common terms. Let's examine the first expression:

• $\frac{x+10}{(x+9)(x-2)}$

We can cancel the common term $(x+10)$ from the numerator and denominator.

1 / ((x+9)(x-2))

Similarly, for the second expression:

• $\frac{3(x^2-4x+4)}{3(x+10)}$

We can cancel the common term $3$ from the numerator and denominator.

(x^2-4x+4) / (x+10)

Multiplying the Simplified Expressions

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Now that we have simplified the expressions, we can multiply them together:

1 / ((x+9)(x-2)) * (x^2-4x+4) / (x+10)

We can cancel the common term $(x+10)$ from the numerator and denominator.

(x^2-4x+4) / ((x+9)(x-2))

Final Answer

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The final answer is $\boxed{\frac{x^2-4x+4}{(x+9)(x-2)}}$.

Conclusion

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Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, we can simplify complex expressions and arrive at a final answer. Remember to factor the numerators and denominators, cancel common terms, and multiply the simplified expressions together. With practice and patience, you will become proficient in simplifying algebraic expressions and tackling even the most challenging problems.

Frequently Asked Questions

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Q: What is the simplest form of the expression representing this product?

A: The simplest form of the expression is $\frac{x^2-4x+4}{(x+9)(x-2)}$.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, you need to factor the numerators and denominators, cancel common terms, and multiply the simplified expressions together.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid include failing to factor the numerators and denominators, not canceling common terms, and multiplying the expressions together incorrectly.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through example problems, such as the one presented in this article. You can also try simplifying more complex expressions and challenging problems to improve your skills.

Additional Resources

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For more information on simplifying algebraic expressions, you can consult the following resources:

• Algebraic Expression Simplification
• Simplifying Algebraic Expressions
• Algebraic Expression Simplification Practice Problems

By following the steps outlined in this article and practicing regularly, you will become proficient in simplifying algebraic expressions and tackling even the most challenging problems.

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Introduction

---

Simplifying algebraic expressions is an essential skill for students and professionals alike. In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given problem: $\frac{x+10}{x^2+7x-18} \cdot \frac{3x^2-12x+12}{3x+30}$. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

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Q: What is the simplest form of the expression representing this product?

A: The simplest form of the expression is $\frac{x^2-4x+4}{(x+9)(x-2)}$.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, you need to follow these steps:

1. Factor the numerators and denominators.
2. Cancel common terms.
3. Multiply the simplified expressions together.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid include:

• Failing to factor the numerators and denominators.
• Not canceling common terms.
• Multiplying the expressions together incorrectly.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through example problems, such as the one presented in this article. You can also try simplifying more complex expressions and challenging problems to improve your skills.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications, including:

• Physics: Simplifying algebraic expressions is essential in physics, where complex equations need to be solved to understand the behavior of physical systems.
• Engineering: Simplifying algebraic expressions is crucial in engineering, where complex equations need to be solved to design and optimize systems.
• Economics: Simplifying algebraic expressions is essential in economics, where complex equations need to be solved to understand the behavior of economic systems.

Q: How can I use technology to simplify algebraic expressions?

A: You can use technology, such as calculators and computer software, to simplify algebraic expressions. Some popular tools include:

• Graphing calculators: These calculators can be used to graph and simplify algebraic expressions.
• Computer algebra systems: These systems can be used to simplify algebraic expressions and solve complex equations.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

• Start by factoring the numerators and denominators.
• Cancel common terms carefully.
• Multiply the simplified expressions together correctly.

Conclusion

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Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article and practicing regularly, you will become proficient in simplifying algebraic expressions and tackling even the most challenging problems. Remember to factor the numerators and denominators, cancel common terms, and multiply the simplified expressions together. With practice and patience, you will become a master of simplifying algebraic expressions.

Frequently Asked Questions

---

Q: What is the simplest form of the expression representing this product?

A: The simplest form of the expression is $\frac{x^2-4x+4}{(x+9)(x-2)}$.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, you need to follow these steps:

1. Factor the numerators and denominators.
2. Cancel common terms.
3. Multiply the simplified expressions together.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid include:

• Failing to factor the numerators and denominators.
• Not canceling common terms.
• Multiplying the expressions together incorrectly.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through example problems, such as the one presented in this article. You can also try simplifying more complex expressions and challenging problems to improve your skills.

Additional Resources

---

For more information on simplifying algebraic expressions, you can consult the following resources:

• Algebraic Expression Simplification
• Simplifying Algebraic Expressions
• Algebraic Expression Simplification Practice Problems

By following the steps outlined in this article and practicing regularly, you will become proficient in simplifying algebraic expressions and tackling even the most challenging problems.