Divide: 5 X − 15 3 X 2 + 9 X ÷ X − 3 3 X 2 \frac{5x - 15}{3x^2 + 9x} \div \frac{x - 3}{3x^2} 3 X 2 + 9 X 5 X − 15 ​ ÷ 3 X 2 X − 3 ​

$$

Introduction to Division of Fractions

Division of fractions is a fundamental concept in mathematics that involves dividing one fraction by another. It is a crucial operation in algebra and is used extensively in various mathematical applications. In this article, we will focus on dividing the given expression: $\frac{5x - 15}{3x^2 + 9x} \div \frac{x - 3}{3x^2}$. We will break down the problem step by step and provide a clear explanation of the process involved.

Understanding the Problem

The given expression involves the division of two fractions. To divide fractions, we need to follow a specific procedure. The first step is to invert the second fraction, which means we need to flip the numerator and denominator. In this case, the second fraction becomes $\frac{3x^2}{x - 3}$. The next step is to multiply the first fraction by the inverted second fraction.

Inverting the Second Fraction

To invert the second fraction, we need to flip the numerator and denominator. The second fraction becomes $\frac{3x^2}{x - 3}$. This is a crucial step in the division of fractions, as it allows us to multiply the fractions together.

Multiplying the Fractions

Now that we have inverted the second fraction, we can multiply the two fractions together. To do this, we need to multiply the numerators and denominators separately. The numerator of the first fraction is $5x - 15$, and the numerator of the inverted second fraction is $3x^2$. Multiplying these two numerators together gives us $(5x - 15) \cdot 3x^2 = 15x^3 - 45x^2$. The denominator of the first fraction is $3x^2 + 9x$, and the denominator of the inverted second fraction is $x - 3$. Multiplying these two denominators together gives us $(3x^2 + 9x) \cdot (x - 3) = 3x^3 - 9x^2 + 27x^2 - 81x = 18x^2 - 81x$.

Simplifying the Expression

Now that we have multiplied the fractions together, we can simplify the expression. To do this, we need to factor out any common factors from the numerator and denominator. The numerator is $15x^3 - 45x^2$, and the denominator is $18x^2 - 81x$. Factoring out $3x^2$ from the numerator gives us $3x^2(5x - 15)$. Factoring out $9x$ from the denominator gives us $9x(2x - 9)$.

Canceling Common Factors

Now that we have factored out common factors from the numerator and denominator, we can cancel them out. The numerator is $3x^2(5x - 15)$, and the denominator is $9x(2x - 9)$. We can cancel out the $3x^2$ from the numerator and the $9x$ from the denominator. This gives us $\frac{5x - 15}{2x - 9}$.

Final Answer

The final answer to the given expression is $\frac{5x - 15}{2x - 9}$. This is the result of dividing the two fractions together and simplifying the expression.

Conclusion

In this article, we have discussed the division of fractions and provided a step-by-step solution to the given expression. We have shown how to invert the second fraction, multiply the fractions together, simplify the expression, and cancel out common factors. The final answer is $\frac{5x - 15}{2x - 9}$.

Frequently Asked Questions

• What is the division of fractions? The division of fractions is a mathematical operation that involves dividing one fraction by another.
• How do I divide fractions? To divide fractions, you need to invert the second fraction and multiply the first fraction by the inverted second fraction.
• What is the final answer to the given expression? The final answer to the given expression is $\frac{5x - 15}{2x - 9}$.

Further Reading

• Division of Fractions: A Comprehensive Guide This article provides a comprehensive guide to the division of fractions, including examples and exercises.
• Algebra: A Beginner's Guide This book provides a beginner's guide to algebra, including the division of fractions.
• Mathematics: A Subject Guide This subject guide provides an overview of mathematics, including the division of fractions.

References

• Algebra: A Comprehensive Guide This book provides a comprehensive guide to algebra, including the division of fractions.
• Mathematics: A Subject Guide This subject guide provides an overview of mathematics, including the division of fractions.

Glossary

• Division of Fractions: A mathematical operation that involves dividing one fraction by another.
• Inverting a Fraction: Flipping the numerator and denominator of a fraction.
• Multiplying Fractions: Multiplying the numerators and denominators of two fractions together.
• Simplifying an Expression: Factoring out common factors from the numerator and denominator of an expression and canceling them out.
  

Introduction

The division of fractions is a fundamental concept in mathematics that involves dividing one fraction by another. It is a crucial operation in algebra and is used extensively in various mathematical applications. In this article, we will provide answers to frequently asked questions related to the division of fractions.

Q&A

Q: What is the division of fractions?

A: The division of fractions is a mathematical operation that involves dividing one fraction by another.

Q: How do I divide fractions?

A: To divide fractions, you need to invert the second fraction and multiply the first fraction by the inverted second fraction.

Q: What is the rule for dividing fractions?

A: The rule for dividing fractions is to invert the second fraction and multiply the first fraction by the inverted second fraction.

Q: Can I divide a fraction by a whole number?

A: Yes, you can divide a fraction by a whole number. To do this, you need to multiply the fraction by the reciprocal of the whole number.

Q: Can I divide a whole number by a fraction?

A: Yes, you can divide a whole number by a fraction. To do this, you need to multiply the whole number by the reciprocal of the fraction.

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

A: Dividing fractions involves inverting the second fraction and multiplying the first fraction by the inverted second fraction. Multiplying fractions involves multiplying the numerators and denominators of the two fractions together.

Q: Can I simplify a fraction after dividing it?

A: Yes, you can simplify a fraction after dividing it. To do this, you need to factor out common factors from the numerator and denominator and cancel them out.

Q: What is the final answer to the given expression: $\frac{5x - 15}{3x^2 + 9x} \div \frac{x - 3}{3x^2}$?

A: The final answer to the given expression is $\frac{5x - 15}{2x - 9}$.

Q: How do I know if a fraction is in its simplest form?

A: A fraction is in its simplest form if there are no common factors between the numerator and denominator.

Q: Can I divide a fraction by a fraction with a zero denominator?

A: No, you cannot divide a fraction by a fraction with a zero denominator. This is because division by zero is undefined.

Q: Can I divide a fraction by a fraction with a negative denominator?

A: Yes, you can divide a fraction by a fraction with a negative denominator. To do this, you need to follow the same rules as dividing fractions with positive denominators.

Conclusion

In this article, we have provided answers to frequently asked questions related to the division of fractions. We have covered topics such as the rule for dividing fractions, dividing fractions by whole numbers, and simplifying fractions after dividing. We hope that this article has been helpful in clarifying any confusion you may have had about the division of fractions.

Further Reading

• Division of Fractions: A Comprehensive Guide This article provides a comprehensive guide to the division of fractions, including examples and exercises.
• Algebra: A Beginner's Guide This book provides a beginner's guide to algebra, including the division of fractions.
• Mathematics: A Subject Guide This subject guide provides an overview of mathematics, including the division of fractions.

References

• Algebra: A Comprehensive Guide This book provides a comprehensive guide to algebra, including the division of fractions.
• Mathematics: A Subject Guide This subject guide provides an overview of mathematics, including the division of fractions.

Glossary

• Division of Fractions: A mathematical operation that involves dividing one fraction by another.
• Inverting a Fraction: Flipping the numerator and denominator of a fraction.
• Multiplying Fractions: Multiplying the numerators and denominators of two fractions together.
• Simplifying an Expression: Factoring out common factors from the numerator and denominator of an expression and canceling them out.