Given Rational Expressions

Oct 13, 2025 by miodeya.com 67 views

Introduction

In mathematics, rational expressions are a fundamental concept in algebra. They are fractions that contain variables in the numerator and/or denominator. When we rewrite rational expressions, we aim to simplify them by factoring the denominator and expressing them in a more manageable form. In this article, we will focus on rewriting two given rational expressions with the denominator ${(3y+5)(y+2)(y+1)}$ .

Rational Expressions

A rational expression is a fraction that contains variables in the numerator and/or denominator. It can be written in the form:

\frac{p(x)}{q(x)}

where ${p(x)}$ and ${q(x)}$ are polynomials.

Given Rational Expressions

We are given two rational expressions:

\frac{7}{3y^2+11y+10}, \quad \frac{y}{3y^2+8y+5}

Our goal is to rewrite these expressions with the denominator ${(3y+5)(y+2)(y+1)}$ .

Step 1: Factor the Denominators

To rewrite the rational expressions, we need to factor the denominators. Let's start with the first expression:

\frac{7}{3y^2+11y+10}

We can factor the denominator as follows:

3y^2+11y+10 = (3y+5)(y+2)

Now, let's factor the second expression:

\frac{y}{3y^2+8y+5}

We can factor the denominator as follows:

3y^2+8y+5 = (3y+5)(y+1)

Step 2: Rewrite the Rational Expressions

Now that we have factored the denominators, we can rewrite the rational expressions with the denominator ${(3y+5)(y+2)(y+1)}$ .

For the first expression:

\frac{7}{(3y+5)(y+2)}

We can rewrite it as:

\frac{7(y+1)}{(3y+5)(y+2)(y+1)}

For the second expression:

\frac{y}{(3y+5)(y+1)}

We can rewrite it as:

\frac{y(y+2)}{(3y+5)(y+1)(y+2)}

Simplifying the Rational Expressions

Now that we have rewritten the rational expressions, we can simplify them by canceling out any common factors in the numerator and denominator.

For the first expression:

\frac{7(y+1)}{(3y+5)(y+2)(y+1)}

We can cancel out the common factor ${(y+1)}$ in the numerator and denominator:

\frac{7}{(3y+5)(y+2)}

For the second expression:

\frac{y(y+2)}{(3y+5)(y+1)(y+2)}

We can cancel out the common factor ${(y+2)}$ in the numerator and denominator:

\frac{y}{(3y+5)(y+1)}

Conclusion

In this article, we have rewritten two given rational expressions with the denominator ${(3y+5)(y+2)(y+1)}$ . We started by factoring the denominators and then rewrote the rational expressions with the desired denominator. Finally, we simplified the rational expressions by canceling out any common factors in the numerator and denominator. This process demonstrates the importance of factoring and simplifying rational expressions in mathematics.

Final Answer

The final answer is:

\frac{7}{(3y+5)(y+2)}, \quad \frac{y}{(3y+5)(y+1)}$$<br/> **Rewrite as Equivalent Rational Expressions: Q&A** ============================================= **Introduction** --------------- In our previous article, we explored how to rewrite rational expressions with a specific denominator. We factored the denominators, rewrote the expressions, and simplified them by canceling out common factors. In this article, we will answer some frequently asked questions (FAQs) related to rewriting rational expressions. **Q: What is the purpose of rewriting rational expressions?** -------------------------------------------------------- A: The purpose of rewriting rational expressions is to simplify them by factoring the denominator and expressing them in a more manageable form. This can help us to better understand the behavior of the expression and make it easier to work with. **Q: How do I know when to rewrite a rational expression?** ------------------------------------------------------ A: You should rewrite a rational expression when the denominator is not in its simplest form, or when you need to express the expression in a specific form. For example, if you need to find the zeros of a rational expression, you may need to rewrite it in a factored form. **Q: What are some common mistakes to avoid when rewriting rational expressions?** -------------------------------------------------------------------------------- A: Some common mistakes to avoid when rewriting rational expressions include: * Not factoring the denominator completely * Not canceling out common factors in the numerator and denominator * Not checking for any remaining common factors after simplifying the expression **Q: How do I factor the denominator of a rational expression?** --------------------------------------------------------- A: To factor the denominator of a rational expression, you can use various factoring techniques, such as: * Factoring out a greatest common factor (GCF) * Factoring by grouping * Factoring quadratic expressions **Q: What is the difference between a rational expression and a rational function?** -------------------------------------------------------------------------------- A: A rational expression is a fraction that contains variables in the numerator and/or denominator. A rational function is a function that is defined by a rational expression. In other words, a rational function is a function that can be expressed as a ratio of two polynomials. **Q: Can I rewrite a rational expression with a denominator that is not in its simplest form?** ----------------------------------------------------------------------------------------- A: Yes, you can rewrite a rational expression with a denominator that is not in its simplest form. However, you should be careful to factor the denominator completely and cancel out any common factors in the numerator and denominator. **Q: How do I know if a rational expression is equivalent to another rational expression?** -------------------------------------------------------------------------------------- A: Two rational expressions are equivalent if they have the same value for all values of the variable. To determine if two rational expressions are equivalent, you can simplify them and compare their simplified forms. **Q: Can I rewrite a rational expression with a denominator that has a variable in the denominator?** ----------------------------------------------------------------------------------------- A: Yes, you can rewrite a rational expression with a denominator that has a variable in the denominator. However, you should be careful to factor the denominator completely and cancel out any common factors in the numerator and denominator. **Conclusion** ---------- In this article, we have answered some frequently asked questions related to rewriting rational expressions. We have discussed the purpose of rewriting rational expressions, common mistakes to avoid, and how to factor the denominator of a rational expression. We have also explored the difference between a rational expression and a rational function, and how to determine if two rational expressions are equivalent. By following these guidelines, you can become more confident in your ability to rewrite rational expressions and simplify them. **Final Answer** -------------- The final answer is: * Rewrite rational expressions to simplify them and express them in a more manageable form. * Factor the denominator completely and cancel out any common factors in the numerator and denominator. * Be careful when rewriting rational expressions with denominators that are not in their simplest form. * Use various factoring techniques to factor the denominator of a rational expression. * Compare the simplified forms of two rational expressions to determine if they are equivalent.