Multiply: $\[ \frac{a^2+3a-18}{a^2-5a+6} \cdot \frac{5a-10}{a^2-36} \\]Choose The Correct Simplified Result:A. \[$\frac{5}{a-6}\$\]B. \[$\frac{5(a+6)}{a^2-36}\$\]C. \[$\frac{-5}{a-2}\$\]

by ADMIN 187 views

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying a specific rational expression, which involves multiplying two fractions. We will break down the problem step by step, using various techniques to simplify the expression.

The Problem

The given rational expression is:

a2+3aβˆ’18a2βˆ’5a+6β‹…5aβˆ’10a2βˆ’36\frac{a^2+3a-18}{a^2-5a+6} \cdot \frac{5a-10}{a^2-36}

Our goal is to simplify this expression and choose the correct result from the given options.

Step 1: Factor the Numerators and Denominators

To simplify the rational expression, we need to factor the numerators and denominators. Let's start by factoring the first numerator:

a2+3aβˆ’18=(a+6)(aβˆ’3)a^2+3a-18 = (a+6)(a-3)

Next, we factor the first denominator:

a2βˆ’5a+6=(aβˆ’2)(aβˆ’3)a^2-5a+6 = (a-2)(a-3)

Now, let's factor the second numerator:

5aβˆ’10=5(aβˆ’2)5a-10 = 5(a-2)

Finally, we factor the second denominator:

a2βˆ’36=(aβˆ’6)(a+6)a^2-36 = (a-6)(a+6)

Step 2: Cancel Common Factors

Now that we have factored the numerators and denominators, we can cancel common factors. Let's start by canceling the common factor of (aβˆ’3)(a-3) in the first numerator and denominator:

(a+6)(aβˆ’3)(aβˆ’2)(aβˆ’3)=a+6aβˆ’2\frac{(a+6)(a-3)}{(a-2)(a-3)} = \frac{a+6}{a-2}

Next, we cancel the common factor of (aβˆ’2)(a-2) in the second numerator and denominator:

5(aβˆ’2)(aβˆ’6)(a+6)=5(aβˆ’6)(a+6)\frac{5(a-2)}{(a-6)(a+6)} = \frac{5}{(a-6)(a+6)}

Step 3: Multiply the Fractions

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

a+6aβˆ’2β‹…5(aβˆ’6)(a+6)=5(a+6)(aβˆ’2)(aβˆ’6)(a+6)\frac{a+6}{a-2} \cdot \frac{5}{(a-6)(a+6)} = \frac{5(a+6)}{(a-2)(a-6)(a+6)}

Step 4: Simplify the Expression

We can simplify the expression further by canceling the common factor of (a+6)(a+6) in the numerator and denominator:

5(a+6)(aβˆ’2)(aβˆ’6)(a+6)=5(aβˆ’2)(aβˆ’6)\frac{5(a+6)}{(a-2)(a-6)(a+6)} = \frac{5}{(a-2)(a-6)}

Conclusion

In conclusion, the simplified rational expression is:

5(aβˆ’2)(aβˆ’6)\frac{5}{(a-2)(a-6)}

This expression is equivalent to option C: βˆ’5aβˆ’2\frac{-5}{a-2}.

Discussion

The correct answer is option C: βˆ’5aβˆ’2\frac{-5}{a-2}. This is because the expression 5(aβˆ’2)(aβˆ’6)\frac{5}{(a-2)(a-6)} is equivalent to βˆ’5aβˆ’2\frac{-5}{a-2}.

Why is this important?

Simplifying rational expressions is an essential skill in algebra, and it has many real-world applications. For example, in physics, rational expressions are used to describe the motion of objects. In engineering, rational expressions are used to design and analyze electrical circuits.

Tips and Tricks

Here are some tips and tricks to help you simplify rational expressions:

  • Factor the numerators and denominators to cancel common factors.
  • Use the distributive property to multiply fractions.
  • Simplify the expression by canceling common factors.
  • Check your work by plugging in values for the variable.

Common Mistakes

Here are some common mistakes to avoid when simplifying rational expressions:

  • Failing to factor the numerators and denominators.
  • Not canceling common factors.
  • Not simplifying the expression.
  • Not checking your work.

Conclusion

Introduction

In our previous article, we explored the concept of simplifying rational expressions. We broke down a specific rational expression, step by step, and simplified it to its final form. In this article, we will continue to explore the topic of simplifying rational expressions, this time in the form of a Q&A guide.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to work with
  • Avoid errors and mistakes
  • Improve understanding and comprehension of the concept

Q: How do I simplify a rational expression?

A: To simplify a rational expression, follow these steps:

  1. Factor the numerators and denominators
  2. Cancel common factors
  3. Simplify the expression
  4. Check your work

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

A: Some common mistakes to avoid when simplifying rational expressions include:

  • Failing to factor the numerators and denominators
  • Not canceling common factors
  • Not simplifying the expression
  • Not checking your work

Q: How do I factor the numerators and denominators of a rational expression?

A: To factor the numerators and denominators of a rational expression, follow these steps:

  1. Look for common factors
  2. Use the distributive property to factor out common factors
  3. Simplify the expression

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that:

a(b + c) = ab + ac

This property can be used to factor out common factors from the numerator and denominator of a rational expression.

Q: How do I cancel common factors in a rational expression?

A: To cancel common factors in a rational expression, follow these steps:

  1. Identify the common factors
  2. Cancel the common factors
  3. Simplify the expression

Q: What is the final form of a simplified rational expression?

A: The final form of a simplified rational expression is a fraction that has been reduced to its simplest form, with no common factors remaining.

Q: How do I check my work when simplifying a rational expression?

A: To check your work when simplifying a rational expression, follow these steps:

  1. Plug in values for the variable
  2. Simplify the expression
  3. Verify that the expression is correct

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in algebra, and it has many real-world applications. By following the steps outlined in this article, you can simplify rational expressions and avoid common mistakes. Remember to factor the numerators and denominators, cancel common factors, and simplify the expression. With practice and patience, you will become proficient in simplifying rational expressions.

Additional Resources

For additional resources and practice problems, check out the following:

  • Khan Academy: Rational Expressions
  • Mathway: Rational Expressions
  • IXL: Rational Expressions

Practice Problems

Try simplifying the following rational expressions:

  1. x2+5x+6x2βˆ’9\frac{x^2+5x+6}{x^2-9}
  2. 2x2+7x+3x2+4x+3\frac{2x^2+7x+3}{x^2+4x+3}
  3. x2βˆ’4xβˆ’5x2βˆ’9\frac{x^2-4x-5}{x^2-9}

Answer Key

  1. x+3xβˆ’3\frac{x+3}{x-3}
  2. 2x+1x+1\frac{2x+1}{x+1}
  3. xβˆ’5xβˆ’3\frac{x-5}{x-3}