Fill In The Blank To Make Equivalent Rational Expressions.$\frac{2}{w+5}=\frac{\square}{(w-1)(w+5)}$

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Introduction

In algebra, equivalent rational expressions are expressions that have the same value for all values of the variable. To make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same factor. In this article, we will learn how to fill in the blank to make equivalent rational expressions.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator. For example, the rational expression 2xx\frac{2x}{x} can be simplified to 22 by canceling out the common factor xx.

The Concept of Equivalent Rational Expressions

Equivalent rational expressions are expressions that have the same value for all values of the variable. To make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same factor. This factor is called the equivalent rational expression factor.

How to Fill in the Blank to Make Equivalent Rational Expressions

To fill in the blank to make equivalent rational expressions, we need to follow these steps:

  1. Identify the equivalent rational expression factor: The equivalent rational expression factor is the factor that we need to multiply both the numerator and denominator of one expression by to make the two expressions equivalent.
  2. Multiply the numerator and denominator of one expression by the equivalent rational expression factor: Once we have identified the equivalent rational expression factor, we need to multiply both the numerator and denominator of one expression by this factor.
  3. Simplify the expression: After multiplying the numerator and denominator of one expression by the equivalent rational expression factor, we need to simplify the expression by canceling out any common factors in the numerator and denominator.

Example 1: Filling in the Blank to Make Equivalent Rational Expressions

Let's consider the following example:

2w+5=β–‘(wβˆ’1)(w+5)\frac{2}{w+5}=\frac{\square}{(w-1)(w+5)}

To fill in the blank, we need to identify the equivalent rational expression factor. In this case, the equivalent rational expression factor is (wβˆ’1)(w-1).

We can multiply the numerator and denominator of the left-hand side expression by (wβˆ’1)(w-1) to make the two expressions equivalent:

2w+5=2(wβˆ’1)(w+5)(wβˆ’1)\frac{2}{w+5}=\frac{2(w-1)}{(w+5)(w-1)}

Now, we can simplify the expression by canceling out the common factor (w+5)(w+5) in the numerator and denominator:

2(wβˆ’1)(w+5)(wβˆ’1)=2(wβˆ’1)w2+4wβˆ’5\frac{2(w-1)}{(w+5)(w-1)}=\frac{2(w-1)}{w^2+4w-5}

Therefore, the equivalent rational expression is 2(wβˆ’1)w2+4wβˆ’5\frac{2(w-1)}{w^2+4w-5}.

Example 2: Filling in the Blank to Make Equivalent Rational Expressions

Let's consider another example:

3wβˆ’2=β–‘(w+3)(wβˆ’2)\frac{3}{w-2}=\frac{\square}{(w+3)(w-2)}

To fill in the blank, we need to identify the equivalent rational expression factor. In this case, the equivalent rational expression factor is (w+3)(w+3).

We can multiply the numerator and denominator of the left-hand side expression by (w+3)(w+3) to make the two expressions equivalent:

3wβˆ’2=3(w+3)(wβˆ’2)(w+3)\frac{3}{w-2}=\frac{3(w+3)}{(w-2)(w+3)}

Now, we can simplify the expression by canceling out the common factor (wβˆ’2)(w-2) in the numerator and denominator:

3(w+3)(wβˆ’2)(w+3)=3(w+3)w2+wβˆ’6\frac{3(w+3)}{(w-2)(w+3)}=\frac{3(w+3)}{w^2+ w-6}

Therefore, the equivalent rational expression is 3(w+3)w2+wβˆ’6\frac{3(w+3)}{w^2+ w-6}.

Conclusion

In conclusion, filling in the blank to make equivalent rational expressions involves identifying the equivalent rational expression factor and multiplying both the numerator and denominator of one expression by this factor. We can then simplify the expression by canceling out any common factors in the numerator and denominator. By following these steps, we can make two rational expressions equivalent.

Frequently Asked Questions

Q: What is the equivalent rational expression factor?

A: The equivalent rational expression factor is the factor that we need to multiply both the numerator and denominator of one expression by to make the two expressions equivalent.

Q: How do I identify the equivalent rational expression factor?

A: To identify the equivalent rational expression factor, we need to look for the factor that is common to both the numerator and denominator of the two expressions.

Q: What is the purpose of multiplying the numerator and denominator of one expression by the equivalent rational expression factor?

A: The purpose of multiplying the numerator and denominator of one expression by the equivalent rational expression factor is to make the two expressions equivalent.

Q: How do I simplify the expression after multiplying the numerator and denominator of one expression by the equivalent rational expression factor?

A: To simplify the expression, we need to cancel out any common factors in the numerator and denominator.

Final Thoughts

In this article, we have learned how to fill in the blank to make equivalent rational expressions. We have also discussed the concept of equivalent rational expressions and how to identify the equivalent rational expression factor. By following the steps outlined in this article, we can make two rational expressions equivalent.

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Introduction

In our previous article, we discussed how to fill in the blank to make equivalent rational expressions. However, we know that there are many more questions that our readers may have. In this article, we will answer some of the most frequently asked questions about fill in the blank to make equivalent rational expressions.

Q&A

Q: What is the equivalent rational expression factor?

A: The equivalent rational expression factor is the factor that we need to multiply both the numerator and denominator of one expression by to make the two expressions equivalent.

Q: How do I identify the equivalent rational expression factor?

A: To identify the equivalent rational expression factor, we need to look for the factor that is common to both the numerator and denominator of the two expressions.

Q: What is the purpose of multiplying the numerator and denominator of one expression by the equivalent rational expression factor?

A: The purpose of multiplying the numerator and denominator of one expression by the equivalent rational expression factor is to make the two expressions equivalent.

Q: How do I simplify the expression after multiplying the numerator and denominator of one expression by the equivalent rational expression factor?

A: To simplify the expression, we need to cancel out any common factors in the numerator and denominator.

Q: Can I use any factor as the equivalent rational expression factor?

A: No, we can only use a factor that is common to both the numerator and denominator of the two expressions as the equivalent rational expression factor.

Q: What if the equivalent rational expression factor is a fraction?

A: If the equivalent rational expression factor is a fraction, we need to multiply both the numerator and denominator of one expression by the numerator and denominator of the fraction, respectively.

Q: Can I use a variable as the equivalent rational expression factor?

A: Yes, we can use a variable as the equivalent rational expression factor, but we need to make sure that the variable is not already present in the expression.

Q: How do I know if the two expressions are equivalent?

A: To know if the two expressions are equivalent, we need to check if they have the same value for all values of the variable.

Q: Can I use a calculator to check if the two expressions are equivalent?

A: Yes, we can use a calculator to check if the two expressions are equivalent, but we need to make sure that the calculator is set to the correct mode and that we are using the correct values.

Examples

Example 1: Identifying the Equivalent Rational Expression Factor

Let's consider the following example:

2w+5=β–‘(wβˆ’1)(w+5)\frac{2}{w+5}=\frac{\square}{(w-1)(w+5)}

To identify the equivalent rational expression factor, we need to look for the factor that is common to both the numerator and denominator of the two expressions. In this case, the equivalent rational expression factor is (wβˆ’1)(w-1).

Example 2: Multiplying the Numerator and Denominator by the Equivalent Rational Expression Factor

Let's consider the following example:

3wβˆ’2=β–‘(w+3)(wβˆ’2)\frac{3}{w-2}=\frac{\square}{(w+3)(w-2)}

To multiply the numerator and denominator of one expression by the equivalent rational expression factor, we need to multiply both the numerator and denominator of the left-hand side expression by (w+3)(w+3):

3wβˆ’2=3(w+3)(wβˆ’2)(w+3)\frac{3}{w-2}=\frac{3(w+3)}{(w-2)(w+3)}

Example 3: Simplifying the Expression

Let's consider the following example:

2(wβˆ’1)(w+5)(wβˆ’1)=2(wβˆ’1)w2+4wβˆ’5\frac{2(w-1)}{(w+5)(w-1)}=\frac{2(w-1)}{w^2+4w-5}

To simplify the expression, we need to cancel out any common factors in the numerator and denominator. In this case, we can cancel out the common factor (wβˆ’1)(w-1) in the numerator and denominator:

2(wβˆ’1)(w+5)(wβˆ’1)=2w+5\frac{2(w-1)}{(w+5)(w-1)}=\frac{2}{w+5}

Conclusion

In conclusion, fill in the blank to make equivalent rational expressions involves identifying the equivalent rational expression factor and multiplying both the numerator and denominator of one expression by this factor. We can then simplify the expression by canceling out any common factors in the numerator and denominator. By following these steps, we can make two rational expressions equivalent.

Final Thoughts

In this article, we have answered some of the most frequently asked questions about fill in the blank to make equivalent rational expressions. We hope that this article has been helpful in clarifying any doubts that our readers may have had. If you have any further questions, please don't hesitate to contact us.