Divide And Simplify.$\[ \frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14} \\]

Introduction

When it comes to algebraic division, one of the most common challenges students face is simplifying complex fractions. In this article, we will delve into the world of algebraic division and explore the concept of dividing and simplifying fractions. We will use the given example, $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$, to demonstrate the step-by-step process of dividing and simplifying fractions.

Understanding the Concept of Division

Before we dive into the example, let's take a moment to understand the concept of division. Division is the process of finding the quotient of two numbers or expressions. In the context of algebraic division, we are dealing with fractions, which are a type of expression that represents a part of a whole. When we divide one fraction by another, we are essentially finding the quotient of the two fractions.

The Given Example

The given example is $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Factor the Denominators

To simplify the given expression, we need to factor the denominators of both fractions. The first fraction has a denominator of $y^2-4$, which can be factored as $(y+2)(y-2)$. The second fraction has a denominator of $y^2+5y-14$, which can be factored as $(y+7)(y-2)$.

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression as:

$$\frac{y^3+2y}{(y+2)(y-2)} \div \frac{(y+7)(y-2)}{y^2+5y-14}$$

Step 3: Invert and Multiply

To divide fractions, we need to invert the second fraction and multiply. Inverting the second fraction means flipping the numerator and denominator, so we get:

$$\frac{y^3+2y}{(y+2)(y-2)} \times \frac{y^2+5y-14}{(y+7)(y-2)}$$

Step 4: Cancel Common Factors

Now that we have multiplied the fractions, we can cancel common factors. The expression can be simplified by canceling the common factor of $(y-2)$ in the numerator and denominator.

Step 5: Simplify the Expression

After canceling the common factor, we are left with:

$$\frac{y^3+2y}{(y+2)} \times \frac{y^2+5y-14}{(y+7)}$$

Step 6: Multiply the Numerators and Denominators

Now that we have simplified the expression, we can multiply the numerators and denominators.

Step 7: Simplify the Result

After multiplying the numerators and denominators, we get:

$$\frac{(y^3+2y)(y^2+5y-14)}{(y+2)(y+7)}$$

Step 8: Expand and Simplify

Finally, we can expand and simplify the expression to get the final result.

Conclusion

In this article, we have demonstrated the step-by-step process of dividing and simplifying fractions. We used the given example, $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$, to show how to factor the denominators, invert and multiply, cancel common factors, and simplify the expression. By following these steps, we can simplify even the most complex fractions and arrive at the final result.

Frequently Asked Questions

• Q: What is the order of operations in algebraic division? A: The order of operations in algebraic division is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I factor the denominators of a fraction? A: To factor the denominators of a fraction, look for common factors and use the distributive property to factor the expression.
• Q: How do I invert and multiply fractions? A: To invert and multiply fractions, flip the numerator and denominator of the second fraction and multiply the numerators and denominators.
• Q: How do I cancel common factors in a fraction? A: To cancel common factors in a fraction, look for common factors in the numerator and denominator and cancel them out.

Final Thoughts

Dividing and simplifying fractions can be a challenging task, but with practice and patience, it can become second nature. By following the steps outlined in this article, you can simplify even the most complex fractions and arrive at the final result. Remember to always follow the order of operations and to factor the denominators, invert and multiply, cancel common factors, and simplify the expression. With these skills, you will be well on your way to becoming a master of algebraic division.

Introduction

In our previous article, "Divide and Simplify: A Comprehensive Guide to Algebraic Division," we explored the concept of dividing and simplifying fractions. We used the example, $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$, to demonstrate the step-by-step process of dividing and simplifying fractions. In this article, we will answer some of the most frequently asked questions about dividing and simplifying fractions.

Q&A

Q: What is the order of operations in algebraic division?

A: The order of operations in algebraic division is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I factor the denominators of a fraction?

A: To factor the denominators of a fraction, look for common factors and use the distributive property to factor the expression. For example, the denominator $y^2-4$ can be factored as $(y+2)(y-2)$.

Q: How do I invert and multiply fractions?

A: To invert and multiply fractions, flip the numerator and denominator of the second fraction and multiply the numerators and denominators. For example, to divide $\frac{y^3+2y}{y^2-4}$ by $\frac{y^2+y-42}{y^2+5y-14}$, we would invert the second fraction and multiply: $\frac{y^3+2y}{(y+2)(y-2)} \times \frac{(y+7)(y-2)}{y^2+5y-14}$.

Q: How do I cancel common factors in a fraction?

A: To cancel common factors in a fraction, look for common factors in the numerator and denominator and cancel them out. For example, in the expression $\frac{(y^3+2y)(y^2+5y-14)}{(y+2)(y+7)}$, we can cancel the common factor of $(y-2)$ in the numerator and denominator.

Q: What is the difference between dividing and simplifying fractions?

A: Dividing fractions involves finding the quotient of two fractions, while simplifying fractions involves reducing the fraction to its simplest form. For example, the expression $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$ involves both dividing and simplifying fractions.

Q: How do I know when to simplify a fraction?

A: You should simplify a fraction whenever possible. Simplifying fractions can help to make the expression more manageable and easier to work with.

Q: Can I simplify a fraction that has a variable in the denominator?

A: Yes, you can simplify a fraction that has a variable in the denominator. However, you must be careful to avoid dividing by zero. For example, the expression $\frac{y^3+2y}{y^2-4}$ can be simplified by factoring the denominator and canceling common factors.

Q: How do I know when to use the distributive property to factor an expression?

A: You should use the distributive property to factor an expression whenever possible. The distributive property can help to simplify the expression and make it easier to work with.

Q: Can I use the distributive property to factor a fraction?

A: Yes, you can use the distributive property to factor a fraction. However, you must be careful to avoid dividing by zero. For example, the expression $\frac{y^3+2y}{y^2-4}$ can be factored using the distributive property.

Conclusion

Dividing and simplifying fractions can be a challenging task, but with practice and patience, it can become second nature. By following the steps outlined in this article and answering the frequently asked questions, you can simplify even the most complex fractions and arrive at the final result. Remember to always follow the order of operations and to factor the denominators, invert and multiply, cancel common factors, and simplify the expression. With these skills, you will be well on your way to becoming a master of algebraic division.

Final Thoughts

Dividing and simplifying fractions is an essential skill in algebra, and it requires practice and patience to master. By following the steps outlined in this article and answering the frequently asked questions, you can simplify even the most complex fractions and arrive at the final result. Remember to always follow the order of operations and to factor the denominators, invert and multiply, cancel common factors, and simplify the expression. With these skills, you will be well on your way to becoming a master of algebraic division.

Additional Resources

• Khan Academy: Algebraic Division
• Mathway: Algebraic Division
• Wolfram Alpha: Algebraic Division

Final Exam

Test your skills by taking the final exam. The exam consists of 10 questions that cover the material outlined in this article.

1. What is the order of operations in algebraic division? a) PEMDAS b) PMDAS c) PEMDAS d) PMDAS

Answer: a) PEMDAS

2. How do I factor the denominators of a fraction? a) Look for common factors and use the distributive property to factor the expression. b) Look for common factors and use the distributive property to factor the numerator. c) Look for common factors and use the distributive property to factor the denominator. d) Look for common factors and use the distributive property to factor the expression.

Answer: a) Look for common factors and use the distributive property to factor the expression.

3. How do I invert and multiply fractions? a) Flip the numerator and denominator of the second fraction and multiply the numerators and denominators. b) Flip the numerator and denominator of the first fraction and multiply the numerators and denominators. c) Flip the numerator and denominator of the second fraction and multiply the numerators and denominators. d) Flip the numerator and denominator of the first fraction and multiply the numerators and denominators.

Answer: a) Flip the numerator and denominator of the second fraction and multiply the numerators and denominators.

4. How do I cancel common factors in a fraction? a) Look for common factors in the numerator and denominator and cancel them out. b) Look for common factors in the numerator and denominator and do not cancel them out. c) Look for common factors in the numerator and denominator and cancel them out. d) Look for common factors in the numerator and denominator and do not cancel them out.

Answer: a) Look for common factors in the numerator and denominator and cancel them out.

5. What is the difference between dividing and simplifying fractions? a) Dividing fractions involves finding the quotient of two fractions, while simplifying fractions involves reducing the fraction to its simplest form. b) Dividing fractions involves reducing the fraction to its simplest form, while simplifying fractions involves finding the quotient of two fractions. c) Dividing fractions involves finding the quotient of two fractions, while simplifying fractions involves reducing the fraction to its simplest form. d) Dividing fractions involves reducing the fraction to its simplest form, while simplifying fractions involves finding the quotient of two fractions.

Answer: a) Dividing fractions involves finding the quotient of two fractions, while simplifying fractions involves reducing the fraction to its simplest form.

6. How do I know when to simplify a fraction? a) You should simplify a fraction whenever possible. b) You should simplify a fraction only when necessary. c) You should simplify a fraction only when the denominator is a variable. d) You should simplify a fraction only when the numerator is a variable.

Answer: a) You should simplify a fraction whenever possible.

7. Can I simplify a fraction that has a variable in the denominator? a) Yes, you can simplify a fraction that has a variable in the denominator. b) No, you cannot simplify a fraction that has a variable in the denominator. c) Yes, you can simplify a fraction that has a variable in the denominator, but you must be careful to avoid dividing by zero. d) No, you cannot simplify a fraction that has a variable in the denominator, but you can simplify it if the variable is a constant.

Answer: a) Yes, you can simplify a fraction that has a variable in the denominator.

8. How do I know when to use the distributive property to factor an expression? a) You should use the distributive property to factor an expression whenever possible. b) You should use the distributive property to factor an expression only when necessary. c) You should use the distributive property to factor an expression only when the denominator is a variable. d) You should use the distributive property to factor an expression only when the numerator is a variable.

Answer: a) You should use the distributive property to factor an expression whenever possible.

9. Can I use the distributive property to factor a fraction? a) Yes, you can use the distributive property to factor a fraction. b) No, you cannot use the distributive property to factor a fraction. c) Yes, you can use the distributive property to factor a fraction, but you must be careful to avoid dividing by zero. d) No, you cannot use the distributive property to factor a fraction, but you can simplify it if the variable is a constant.

Answer: a) Yes, you can use the distributive property to factor a fraction.

10. What is the final result of the expression $\frac{y^3+2y}{y^2-4} \div \frac{y^2+y-42}{y^2+5y-14}$? a) $\frac{(y