Divide The Following Expression. Assume That All Expressions Are Defined.$\[ \frac{x^2+x-12}{9x^2-1} \div \frac{x^2-5x+6}{3x^2-5x-2} \\]Options:A. \[$\frac{x+4}{3x-1}\$\]B. \[$\frac{(x+4)(x+3)}{(3x-1)(x-3)}\$\]C.

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Introduction

In algebra, dividing expressions is a crucial operation that helps us simplify complex equations and solve problems. When dividing algebraic expressions, we need to follow a specific set of rules to ensure that we get the correct result. In this article, we will explore the process of dividing algebraic expressions, focusing on the given expression x2+xβˆ’129x2βˆ’1Γ·x2βˆ’5x+63x2βˆ’5xβˆ’2\frac{x^2+x-12}{9x^2-1} \div \frac{x^2-5x+6}{3x^2-5x-2}.

Understanding the Problem

Before we dive into the solution, let's break down the given expression. We have two fractions, x2+xβˆ’129x2βˆ’1\frac{x^2+x-12}{9x^2-1} and x2βˆ’5x+63x2βˆ’5xβˆ’2\frac{x^2-5x+6}{3x^2-5x-2}, which are being divided. To divide fractions, we need to invert the second fraction and multiply it by the first fraction.

Step 1: Factorize the Numerators and Denominators

To simplify the expression, we need to factorize the numerators and denominators of both fractions.

  • The numerator of the first fraction, x2+xβˆ’12x^2+x-12, can be factored as (x+4)(xβˆ’3)(x+4)(x-3).
  • The denominator of the first fraction, 9x2βˆ’19x^2-1, can be factored as (3xβˆ’1)(3x+1)(3x-1)(3x+1).
  • The numerator of the second fraction, x2βˆ’5x+6x^2-5x+6, can be factored as (xβˆ’3)(xβˆ’2)(x-3)(x-2).
  • The denominator of the second fraction, 3x2βˆ’5xβˆ’23x^2-5x-2, can be factored as (3x+1)(xβˆ’2)(3x+1)(x-2).

Step 2: Invert the Second Fraction and Multiply

Now that we have factored the numerators and denominators, we can invert the second fraction and multiply it by the first fraction.

(x+4)(xβˆ’3)(3xβˆ’1)(3x+1)β‹…(3x+1)(xβˆ’2)(xβˆ’3)(xβˆ’2)\frac{(x+4)(x-3)}{(3x-1)(3x+1)} \cdot \frac{(3x+1)(x-2)}{(x-3)(x-2)}

Step 3: Cancel Out Common Factors

When multiplying fractions, we can cancel out common factors in the numerators and denominators.

(x+4)(xβˆ’3)(3xβˆ’1)(3x+1)β‹…(3x+1)(xβˆ’2)(xβˆ’3)(xβˆ’2)=(x+4)(xβˆ’2)(3xβˆ’1)\frac{(x+4)(x-3)}{(3x-1)(3x+1)} \cdot \frac{(3x+1)(x-2)}{(x-3)(x-2)} = \frac{(x+4)(x-2)}{(3x-1)}

Conclusion

In conclusion, the given expression x2+xβˆ’129x2βˆ’1Γ·x2βˆ’5x+63x2βˆ’5xβˆ’2\frac{x^2+x-12}{9x^2-1} \div \frac{x^2-5x+6}{3x^2-5x-2} can be simplified to (x+4)(xβˆ’2)(3xβˆ’1)\frac{(x+4)(x-2)}{(3x-1)}. This result can be further simplified by factoring the numerator as (x+4)(xβˆ’2)(x+4)(x-2).

Final Answer

The final answer is (x+4)(x+3)(3xβˆ’1)(xβˆ’3)\boxed{\frac{(x+4)(x+3)}{(3x-1)(x-3)}}.

Discussion

The given expression can be simplified using the rules of algebraic division. By factoring the numerators and denominators, inverting the second fraction, and multiplying, we can simplify the expression to (x+4)(x+3)(3xβˆ’1)(xβˆ’3)\frac{(x+4)(x+3)}{(3x-1)(x-3)}. This result can be further simplified by factoring the numerator as (x+4)(x+3)(x+4)(x+3).

Common Mistakes

When dividing algebraic expressions, it's essential to follow the correct order of operations. Some common mistakes include:

  • Not factoring the numerators and denominators
  • Not inverting the second fraction
  • Not canceling out common factors

Tips and Tricks

To simplify algebraic expressions, follow these tips and tricks:

  • Factorize the numerators and denominators whenever possible
  • Invert the second fraction when dividing
  • Cancel out common factors to simplify the expression

Introduction

In our previous article, we explored the process of dividing algebraic expressions, focusing on the given expression x2+xβˆ’129x2βˆ’1Γ·x2βˆ’5x+63x2βˆ’5xβˆ’2\frac{x^2+x-12}{9x^2-1} \div \frac{x^2-5x+6}{3x^2-5x-2}. In this article, we will answer some frequently asked questions about dividing algebraic expressions.

Q: What is the first step in dividing algebraic expressions?

A: The first step in dividing algebraic expressions is to factorize the numerators and denominators of both fractions.

Q: Why do we need to factorize the numerators and denominators?

A: Factorizing the numerators and denominators helps us simplify the expression by canceling out common factors.

Q: How do we invert the second fraction when dividing?

A: To invert the second fraction, we need to swap the numerator and denominator.

Q: What is the next step after inverting the second fraction?

A: After inverting the second fraction, we need to multiply it by the first fraction.

Q: Can we cancel out common factors when multiplying fractions?

A: Yes, we can cancel out common factors when multiplying fractions.

Q: What is the final step in dividing algebraic expressions?

A: The final step in dividing algebraic expressions is to simplify the resulting expression by canceling out any remaining common factors.

Q: What are some common mistakes to avoid when dividing algebraic expressions?

A: Some common mistakes to avoid when dividing algebraic expressions include:

  • Not factoring the numerators and denominators
  • Not inverting the second fraction
  • Not canceling out common factors

Q: How can we simplify complex algebraic expressions?

A: To simplify complex algebraic expressions, follow these tips and tricks:

  • Factorize the numerators and denominators whenever possible
  • Invert the second fraction when dividing
  • Cancel out common factors to simplify the expression

Q: What is the importance of dividing algebraic expressions?

A: Dividing algebraic expressions is an essential operation in algebra that helps us simplify complex equations and solve problems.

Q: Can we divide algebraic expressions with variables?

A: Yes, we can divide algebraic expressions with variables.

Q: What is the difference between dividing algebraic expressions and dividing rational expressions?

A: Dividing algebraic expressions involves dividing polynomials, while dividing rational expressions involves dividing fractions.

Conclusion

In conclusion, dividing algebraic expressions is a crucial operation in algebra that helps us simplify complex equations and solve problems. By following the correct order of operations and avoiding common mistakes, we can simplify complex algebraic expressions and solve problems with ease.

Final Tips and Tricks

To simplify algebraic expressions, remember to:

  • Factorize the numerators and denominators whenever possible
  • Invert the second fraction when dividing
  • Cancel out common factors to simplify the expression

By following these tips and tricks, you can become a master of dividing algebraic expressions and solve problems with confidence.