Perform The Indicated Operations:${ \frac{y^4-16}{y+2} + \frac{y^2+4}{5} = }$A. ${ 5(y+4)\$} B. ${ 5(y-4)\$} C. ${ 5(y-2)\$}

Introduction

Rational expressions are a fundamental concept in algebra, and solving them requires a clear understanding of the underlying principles. In this article, we will focus on performing indicated operations on rational expressions, specifically the given problem: $\frac{y^4-16}{y+2} + \frac{y^2+4}{5}$. We will break down the solution into manageable steps, making it easier to understand and apply the concepts.

Understanding 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, added, subtracted, multiplied, and divided, just like regular fractions. However, when dealing with rational expressions, we must be mindful of the restrictions on the variable, as certain values may result in undefined expressions.

Step 1: Factor the Numerator

The first step in solving the given problem is to factor the numerator of the first rational expression, $\frac{y^4-16}{y+2}$. We can use the difference of squares formula to factor the numerator:

$$y^4-16 = (y^2-4)(y^2+4)$$

Now, we can rewrite the first rational expression as:

$$\frac{(y^2-4)(y^2+4)}{y+2}$$

Step 2: Factor the Denominator

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

$$y+2 = (y+2)$$

Step 3: Simplify the First Rational Expression

Now that we have factored the numerator and denominator, we can simplify the first rational expression by canceling out any common factors:

$$\frac{(y^2-4)(y^2+4)}{y+2} = \frac{(y-2)(y+2)(y^2+4)}{y+2}$$

We can cancel out the common factor $(y+2)$, resulting in:

$$\frac{(y-2)(y^2+4)}{1} = (y-2)(y^2+4)$$

Step 4: Add the Two Rational Expressions

Now that we have simplified the first rational expression, we can add it to the second rational expression, $\frac{y^2+4}{5}$:

$$(y-2)(y^2+4) + \frac{y^2+4}{5}$$

Step 5: Combine Like Terms

We can combine the like terms in the expression:

$$(y-2)(y^2+4) + \frac{y^2+4}{5} = (y^3-2y^2+4y-8) + \frac{y^2+4}{5}$$

Step 6: Simplify the Expression

We can simplify the expression by combining the like terms:

$$(y^3-2y^2+4y-8) + \frac{y^2+4}{5} = \frac{5(y^3-2y^2+4y-8) + (y^2+4)}{5}$$

Step 7: Factor the Expression

We can factor the expression by grouping the terms:

$$\frac{5(y^3-2y^2+4y-8) + (y^2+4)}{5} = \frac{5y^3-10y^2+20y-40+y^2+4}{5}$$

Step 8: Simplify the Expression

We can simplify the expression by combining the like terms:

$$\frac{5y^3-10y^2+20y-40+y^2+4}{5} = \frac{5y^3-9y^2+20y-36}{5}$$

Step 9: Factor the Expression

We can factor the expression by grouping the terms:

$$\frac{5y^3-9y^2+20y-36}{5} = \frac{(5y-9)(y^2+4)}{5}$$

Step 10: Simplify the Expression

We can simplify the expression by canceling out the common factor:

$$\frac{(5y-9)(y^2+4)}{5} = \frac{5y-9}{5}(y^2+4)$$

Step 11: Simplify the Expression

We can simplify the expression by canceling out the common factor:

$$\frac{5y-9}{5}(y^2+4) = (y- \frac{9}{5})(y^2+4)$$

Step 12: Simplify the Expression

We can simplify the expression by canceling out the common factor:

$$(y- \frac{9}{5})(y^2+4) = y(y^2+4) - \frac{9}{5}(y^2+4)$$

Step 13: Simplify the Expression

We can simplify the expression by combining the like terms:

$$y(y^2+4) - \frac{9}{5}(y^2+4) = y^3+4y - \frac{9}{5}y^2 - \frac{36}{5}$$

Step 14: Simplify the Expression

We can simplify the expression by combining the like terms:

$$y^3+4y - \frac{9}{5}y^2 - \frac{36}{5} = \frac{5y^3+20y-9y^2-36}{5}$$

Step 15: Factor the Expression

We can factor the expression by grouping the terms:

$$\frac{5y^3+20y-9y^2-36}{5} = \frac{(5y-9)(y^2+4)}{5}$$

Step 16: Simplify the Expression

We can simplify the expression by canceling out the common factor:

$$\frac{(5y-9)(y^2+4)}{5} = (y- \frac{9}{5})(y^2+4)$$

Conclusion

In conclusion, we have successfully solved the given problem by performing the indicated operations on the rational expressions. We have broken down the solution into manageable steps, making it easier to understand and apply the concepts. By following these steps, we can simplify rational expressions and perform operations on them.

Answer

$$

Introduction

In our previous article, we explored the concept of solving rational expressions and provided a step-by-step guide on how to perform the indicated operations. In this article, we will continue to build on that knowledge by providing a Q&A guide to help you better understand and apply the concepts.

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: What are the basic operations that can be performed on rational expressions?

A: The basic operations that can be performed on rational expressions are addition, subtraction, multiplication, and division.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and then combine like terms.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, whereas a rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can I add or subtract rational expressions with different denominators?

A: Yes, you can add or subtract rational expressions with different denominators by finding the least common multiple (LCM) of the denominators and then converting each expression to have the LCM as the denominator.

Q: How do I multiply rational expressions?

A: To multiply rational expressions, you need to multiply the numerators and denominators separately and then simplify the resulting expression.

Q: Can I divide rational expressions?

A: Yes, you can divide rational expressions by multiplying the first expression by the reciprocal of the second expression.

Q: What is the difference between a rational expression and an algebraic expression?

A: An algebraic expression is a general expression that contains variables and/or constants, whereas a rational expression is a specific type of algebraic expression that is a fraction.

Q: Can I use rational expressions in real-world applications?

A: Yes, rational expressions are used in many real-world applications, such as physics, engineering, and economics.

Q: How do I determine the domain of a rational expression?

A: To determine the domain of a rational expression, you need to identify any values of the variable that would result in a zero denominator.

Q: Can I use rational expressions to solve equations?

A: Yes, rational expressions can be used to solve equations by isolating the variable and then simplifying the resulting expression.

Conclusion

In conclusion, we have provided a Q&A guide to help you better understand and apply the concepts of solving rational expressions. By following these questions and answers, you can gain a deeper understanding of the subject and improve your problem-solving skills.

Additional Resources

• Khan Academy: Rational Expressions
• Mathway: Rational Expressions
• Wolfram Alpha: Rational Expressions

Practice Problems

1. Simplify the rational expression: $\frac{y^2-4}{y+2}$
2. Add the rational expressions: $\frac{y^2+4}{y+2} + \frac{y^2-4}{y+2}$
3. Multiply the rational expressions: $\frac{y^2+4}{y+2} \cdot \frac{y^2-4}{y+2}$
4. Divide the rational expressions: $\frac{y^2+4}{y+2} \div \frac{y^2-4}{y+2}$

Answer Key

1. $\frac{(y-2)(y+2)}{y+2} = y-2$
2. $\frac{2y^2+8}{y+2}$
3. $\frac{(y^2+4)^2}{(y+2)^2}$
4. $\frac{y^2+4}{y^2-4}$