Dividing Rational Expressions What Is The Quotient Of (t+3)/(t+4) ÷ (t^2+7t+12)

Jul 14, 2025 by ADMIN 80 views

What is the Quotient? Solving the Division of Rational Expressions

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In mathematics, understanding how to divide rational expressions is a fundamental skill, especially in algebra. This article will delve into the process of dividing rational expressions, focusing on the specific problem: $(\frac{t+3}{t+4}) ÷ (t^2+7t+12)$ . We will break down the steps, explain the underlying concepts, and arrive at the correct quotient. This exploration will not only provide a solution to this particular problem but also equip you with the knowledge to tackle similar challenges with confidence.

Understanding Rational Expressions

Rational expressions are essentially fractions where the numerator and the denominator are polynomials. Think of them as algebraic fractions. To effectively work with rational expressions, it’s crucial to understand the basic operations such as addition, subtraction, multiplication, and, most importantly for our discussion, division. Each of these operations has specific rules and techniques that must be followed to arrive at the correct answer.

Before we dive into the division problem, let's recap some key concepts. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of polynomials include $t+3$ , $t+4$ , and $t^2+7t+12$ . When we have one polynomial divided by another, we get a rational expression. Simplifying, adding, subtracting, multiplying, or dividing rational expressions often involves factoring polynomials, finding common denominators, and reducing fractions, similar to how we work with numerical fractions. The critical difference is that we are dealing with algebraic expressions rather than numbers, adding a layer of complexity that requires careful attention to detail.

The Division of Rational Expressions: A Step-by-Step Guide

When dividing rational expressions, the core principle is to transform the division problem into a multiplication problem. This is achieved by multiplying the first rational expression by the reciprocal of the second. In other words, dividing by a fraction is the same as multiplying by its inverse. This concept is borrowed directly from the arithmetic of numerical fractions and applies seamlessly to algebraic fractions.

To divide $(\frac{t+3}{t+4})$ by $(t^2+7t+12)$ , we first rewrite the division as multiplication by the reciprocal. The expression $(t^2+7t+12)$ can be considered as a rational expression with a denominator of 1, i.e., $\frac{t^2+7t+12}{1}$ . The reciprocal of this expression is $\frac{1}{t^2+7t+12}$ . Therefore, our division problem transforms into the multiplication problem: $(\frac{t+3}{t+4}) \times (\frac{1}{t^2+7t+12})$ .

Now, the next step is crucial: factoring. Factoring polynomials allows us to simplify expressions and identify common factors that can be canceled out, much like reducing numerical fractions to their simplest form. The polynomial $t^2+7t+12$ can be factored into $(t+3)(t+4)$ . This factorization is a key step in simplifying the expression and revealing the final quotient. By breaking down the quadratic polynomial into its linear factors, we set the stage for canceling out common terms between the numerator and the denominator, which is essential for simplifying the result.

Solving the Problem: $(\frac{t+3}{t+4}) ÷ (t^2+7t+12)$

Let's apply the steps we've discussed to solve the given problem: $(\frac{t+3}{t+4}) ÷ (t^2+7t+12)$ .

Rewrite as multiplication by the reciprocal: As we established, division is equivalent to multiplying by the reciprocal. So, we rewrite the expression as: $(\frac{t+3}{t+4}) \times (\frac{1}{t^2+7t+12})$
Factor the polynomials: The next crucial step is to factor the quadratic polynomial $t^2+7t+12$ . We are looking for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4. Thus, we can factor the polynomial as follows: $t^2+7t+12 = (t+3)(t+4)$ Substituting this factorization into our expression, we get: $(\frac{t+3}{t+4}) \times (\frac{1}{(t+3)(t+4)})$
Multiply the rational expressions: To multiply rational expressions, we multiply the numerators together and the denominators together: $\frac{(t+3) \times 1}{(t+4) \times (t+3)(t+4)} = \frac{t+3}{(t+4)(t+3)(t+4)}$
Simplify by canceling common factors: Now, we look for common factors in the numerator and the denominator that can be canceled out. We see that $(t+3)$ appears in both the numerator and the denominator. Canceling this common factor, we have: $\frac{\cancel{(t+3)}}{(t+4)\cancel{(t+3)}(t+4)} = \frac{1}{(t+4)(t+4)}$
Express the final result: The denominator can be written as $(t+4)^2$ . Therefore, the simplified expression is: $\frac{1}{(t+4)^2}$

Thus, the quotient of $(\frac{t+3}{t+4}) ÷ (t^2+7t+12)$ is $\frac{1}{(t+4)^2}$ .

Analyzing the Answer Choices

Now that we have found the quotient, let's analyze the given answer choices to confirm our result.

The problem presented the following options:

A. $(t+3)^2$ B. $(t+4)^2$ C. $\frac{1}{(t+4)^2}$ D. $\frac{1}{(t+3)^2}$

Our step-by-step solution clearly shows that the correct quotient is $\frac{1}{(t+4)^2}$ , which corresponds to option C. The other options can be ruled out based on our detailed calculation:

Option A, $(t+3)^2$ , is incorrect because it represents the square of $(t+3)$ and does not account for the division and simplification process we performed.
Option B, $(t+4)^2$ , is also incorrect. It represents the square of $(t+4)$ , but it is in the numerator rather than the denominator, and it doesn't reflect the reciprocal and cancellation steps.
Option D, $\frac{1}{(t+3)^2}$ , is incorrect because it has $(t+3)^2$ in the denominator, which is not the result we obtained after simplification. The correct denominator should involve $(t+4)$ terms, not $(t+3)$ terms.

By systematically working through the division, factoring, and simplification steps, we confidently arrived at option C as the correct answer. This methodical approach not only solves the problem but also reinforces the importance of each step in the process of dividing rational expressions.

Common Mistakes to Avoid

When dividing rational expressions, it’s easy to make mistakes if you’re not careful. Identifying these common pitfalls can help prevent errors and ensure you arrive at the correct solution. Here are some frequent mistakes to watch out for:

Forgetting to take the reciprocal: One of the most common errors is forgetting to multiply by the reciprocal of the second rational expression. Remember, division is equivalent to multiplying by the inverse. If you skip this step, you’ll be performing the wrong operation, leading to an incorrect result.
Incorrectly factoring polynomials: Factoring polynomials is a critical step in simplifying rational expressions. Mistakes in factoring, such as incorrect signs or missing factors, can lead to an incorrect quotient. Always double-check your factoring by multiplying the factors back together to ensure they match the original polynomial.
Canceling terms prematurely: Canceling common factors is a key part of simplifying rational expressions, but it must be done correctly. You can only cancel factors that are multiplied, not terms that are added or subtracted. For example, you cannot cancel the $t$ in $\frac{t+3}{t+4}$ because $t$ is a term, not a factor.
Not simplifying completely: After performing the operations, it’s important to simplify the expression as much as possible. This means canceling all common factors and combining like terms. Leaving the expression unsimplified can result in an incorrect answer if the simplified form is required.
Distributing incorrectly: When multiplying polynomials, ensure you distribute correctly. Each term in the first polynomial must be multiplied by each term in the second polynomial. Mistakes in distribution can lead to incorrect expressions and, ultimately, an incorrect quotient.

By being aware of these common mistakes and taking the time to double-check your work, you can minimize errors and improve your accuracy when dividing rational expressions. Attention to detail and a systematic approach are key to success in these types of problems.

Practice Problems

To solidify your understanding of dividing rational expressions, working through practice problems is essential. Here are a few additional problems that are similar to the one we solved in this article:

$\frac{x+2}{x+3} ÷ (x^2+5x+6)$
$\frac{y-1}{y+2} ÷ (y^2+y-2)$
$\frac{2z+1}{z-3} ÷ (4z^2+4z+1)$

These problems offer a range of complexity and will help you apply the steps we’ve discussed in various scenarios. When solving these problems, remember to:

Rewrite the division as multiplication by the reciprocal.
Factor the polynomials.
Multiply the rational expressions.
Simplify by canceling common factors.
Express the final result in its simplest form.

Working through these problems will not only reinforce the process but also help you develop confidence in your ability to divide rational expressions. Each problem presents a unique challenge, and by tackling them systematically, you’ll strengthen your algebraic skills and deepen your understanding of rational expressions.

Conclusion

In conclusion, dividing rational expressions involves several key steps, including rewriting division as multiplication by the reciprocal, factoring polynomials, multiplying the expressions, and simplifying by canceling common factors. By following these steps carefully, we can accurately find the quotient of complex algebraic fractions. In the specific problem $(\frac{t+3}{t+4}) ÷ (t^2+7t+12)$ , we found the quotient to be $\frac{1}{(t+4)^2}$ . This process not only provides the correct answer but also enhances our understanding of algebraic manipulations and simplification techniques.

Understanding the nuances of rational expressions and their operations is crucial for success in algebra and higher-level mathematics. By mastering these concepts, you’ll be well-equipped to tackle more advanced problems and appreciate the elegance of mathematical problem-solving. Remember to practice regularly, pay attention to detail, and review your work to minimize errors. With consistent effort and a solid grasp of the fundamentals, you can confidently navigate the world of rational expressions and algebraic equations.