Determine Which Steps Are Used To Find The Product Shown. Put The Steps In The Order In Which They Would Be Performed.Given Expression:$\[ \frac{x^2+7x+10}{x^2+4x+4} \cdot \frac{x^2+3x+2}{x^2+6x+5} \\]Steps To Find The Product:1. Factor Each

Introduction

In mathematics, rational expressions are a fundamental concept that plays a crucial role in algebra and calculus. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. When we are given a product of two rational expressions, we need to find the product by following a series of steps. In this article, we will discuss the steps to find the product of two rational expressions, using the given expression as an example.

The Given Expression

The given expression is:

$$\frac{x^2+7x+10}{x^2+4x+4} \cdot \frac{x^2+3x+2}{x^2+6x+5}$$

Step 1: Factor Each Expression

The first step in finding the product of two rational expressions is to factor each expression. Factoring involves expressing an expression as a product of simpler expressions, called factors. In this case, we need to factor the numerator and denominator of each rational expression.

Factoring the Numerator and Denominator of the First Rational Expression

The numerator of the first rational expression is $x^2+7x+10$. We can factor this expression as:

$$x^2+7x+10 = (x+5)(x+2)$$

The denominator of the first rational expression is $x^2+4x+4$. We can factor this expression as:

$$x^2+4x+4 = (x+2)^2$$

Factoring the Numerator and Denominator of the Second Rational Expression

The numerator of the second rational expression is $x^2+3x+2$. We can factor this expression as:

$$x^2+3x+2 = (x+1)(x+2)$$

The denominator of the second rational expression is $x^2+6x+5$. We can factor this expression as:

$$x^2+6x+5 = (x+1)(x+5)$$

Step 2: Cancel Out Common Factors

After factoring each expression, we can cancel out common factors between the numerator and denominator of each rational expression. In this case, we can cancel out the common factor $(x+2)$ between the numerator and denominator of the first rational expression, and the common factor $(x+1)$ between the numerator and denominator of the second rational expression.

Canceling Out Common Factors

After canceling out the common factors, the expression becomes:

$$\frac{(x+5)}{(x+2)} \cdot \frac{(x+1)(x+2)}{(x+1)(x+5)}$$

Step 3: Simplify the Expression

The final step in finding the product of two rational expressions is to simplify the expression. In this case, we can simplify the expression by canceling out the common factors between the numerator and denominator.

Simplifying the Expression

After simplifying the expression, we get:

$$\frac{(x+5)}{(x+2)} \cdot \frac{(x+1)(x+2)}{(x+1)(x+5)} = \frac{(x+1)}{(x+2)}$$

Conclusion

In conclusion, the steps to find the product of two rational expressions are:

1. Factor each expression
2. Cancel out common factors
3. Simplify the expression

Introduction

In our previous article, we discussed the steps to find the product of two rational expressions. However, we understand that there may be some questions and doubts that readers may have. In this article, we will address some of the frequently asked questions (FAQs) on finding the product of rational expressions.

Q: What is the first step in finding the product of two rational expressions?

A: The first step in finding the product of two rational expressions is to factor each expression. Factoring involves expressing an expression as a product of simpler expressions, called factors.

Q: How do I factor a rational expression?

A: To factor a rational expression, you need to factor the numerator and denominator separately. You can use the following steps:

1. Factor the numerator and denominator separately.
2. Look for common factors between the numerator and denominator.
3. Cancel out the common factors.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves expressing an expression as a product of simpler expressions, called factors. Canceling out common factors involves removing the common factors between the numerator and denominator.

Q: Can I cancel out common factors between the numerator and denominator if they are not identical?

A: No, you cannot cancel out common factors between the numerator and denominator if they are not identical. The common factors must be identical in order to cancel them out.

Q: What is the final step in finding the product of two rational expressions?

A: The final step in finding the product of two rational expressions is to simplify the expression. This involves canceling out any remaining common factors between the numerator and denominator.

Q: Can I simplify a rational expression by canceling out common factors between the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out common factors between the numerator and denominator. However, you must make sure that the common factors are identical.

Q: What are some common mistakes to avoid when finding the product of rational expressions?

A: Some common mistakes to avoid when finding the product of rational expressions include:

• Not factoring the numerator and denominator separately.
• Not canceling out common factors between the numerator and denominator.
• Canceling out common factors between the numerator and denominator if they are not identical.

Q: How can I practice finding the product of rational expressions?

A: You can practice finding the product of rational expressions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

In conclusion, finding the product of rational expressions involves factoring, canceling out common factors, and simplifying the expression. By following these steps and avoiding common mistakes, you can find the product of rational expressions with ease. If you have any further questions or doubts, please don't hesitate to ask.