Not factoring the numerators and denominators

Oct 13, 2025 by windailys.com 13 views

Introduction

Rational expressions are a fundamental concept in algebra, and multiplying and simplifying them is a crucial skill to master. In this article, we will explore the process of multiplying and simplifying rational expressions, using the given expression as an example.

The Given Expression

The given expression is:

\frac{x^2-6x-7}{2x^2-98} \cdot \frac{x^2+14x+49}{3x^2+24x+21}

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators. Let's start by factoring the first numerator:

x^2-6x-7 = (x-7)(x+1)

Now, let's factor the second numerator:

x^2+14x+49 = (x+7)^2

Next, let's factor the first denominator:

2x^2-98 = 2(x^2-49) = 2(x-7)(x+7)

Finally, let's factor the second denominator:

3x^2+24x+21 = 3(x^2+8x+7) = 3(x+1)(x+7)

Step 2: Rewrite the Expression with Factored Numerators and Denominators

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

\frac{(x-7)(x+1)}{2(x-7)(x+7)} \cdot \frac{(x+7)^2}{3(x+1)(x+7)}

Step 3: Cancel Common Factors

We can now cancel common factors between the numerators and denominators. Let's start by canceling the $(x+7)$ factor in the first numerator and denominator:

\frac{(x-7)(x+1)}{2(x-7)(x+7)} \cdot \frac{(x+7)^2}{3(x+1)(x+7)} = \frac{(x-7)(x+1)}{2(x-7)} \cdot \frac{(x+7)}{3(x+1)}

Next, let's cancel the $(x+1)$ factor in the first numerator and denominator:

\frac{(x-7)(x+1)}{2(x-7)} \cdot \frac{(x+7)}{3(x+1)} = \frac{x-7}{2(x-7)} \cdot \frac{x+7}{3}

Finally, let's cancel the $(x-7)$ factor in the first numerator and denominator:

\frac{x-7}{2(x-7)} \cdot \frac{x+7}{3} = \frac{1}{2} \cdot \frac{x+7}{3}

Step 4: Simplify the Expression

We can now simplify the expression by multiplying the fractions:

\frac{1}{2} \cdot \frac{x+7}{3} = \frac{x+7}{6}

Conclusion

Multiplying and simplifying rational expressions requires careful attention to factoring and canceling common factors. By following the steps outlined in this article, we can simplify even the most complex rational expressions. Remember to always factor the numerators and denominators, cancel common factors, and simplify the expression to its final form.

Common Mistakes to Avoid

When multiplying and simplifying rational expressions, there are several common mistakes to avoid:

Not factoring the numerators and denominators: Failing to factor the numerators and denominators can lead to incorrect simplifications.
Not canceling common factors: Failing to cancel common factors can result in unnecessary complexity.
Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity.

Real-World Applications

Multiplying and simplifying rational expressions has numerous real-world applications, including:

Physics and Engineering: Rational expressions are used to describe the motion of objects and the behavior of electrical circuits.
Economics: Rational expressions are used to model economic systems and make predictions about future trends.
Computer Science: Rational expressions are used to optimize algorithms and solve complex problems.

Practice Problems

To practice multiplying and simplifying rational expressions, try the following problems:

Problem 1: Multiply and simplify the expression $\frac{x^2-4x-5}{x^2+4x+4} \cdot \frac{x^2+4x+4}{x^2-4x-5}$
Problem 2: Multiply and simplify the expression $\frac{x^2+2x-3}{x^2-2x-3} \cdot \frac{x^2-2x-3}{x^2+2x-3}$
Problem 3: Multiply and simplify the expression $\frac{x^2-6x-7}{x^2+6x-7} \cdot \frac{x^2+6x-7}{x^2-6x-7}$

Conclusion

Introduction

Multiplying and simplifying rational expressions is a fundamental concept in algebra, and it can be a challenging topic for many students. In this article, we will provide a Q&A guide to help you understand the process of multiplying and simplifying rational expressions.

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: 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: What is the order of operations when multiplying rational expressions?

A: The order of operations when multiplying rational expressions is:

Multiply the numerators
Multiply the denominators
Simplify the resulting expression

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to:

Factor the numerator and denominator
Cancel common factors
Simplify the resulting expression

Q: What are some common mistakes to avoid when multiplying and simplifying rational expressions?

A: Some common mistakes to avoid when multiplying and simplifying rational expressions include:

Not factoring the numerator and denominator
Not canceling common factors
Not simplifying the resulting expression

Q: How do I know when to cancel common factors?

A: You should cancel common factors when the factors appear in both the numerator and denominator.

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, while a rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

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

A: Yes, you can simplify a rational expression that has a variable in the denominator. However, you need to be careful when canceling common factors, as the variable may not be a common factor.

Q: How do I multiply and simplify rational expressions with exponents?

A: To multiply and simplify rational expressions with exponents, you need to follow the rules of exponentiation, which include:

Multiplying exponents with the same base: a^m * a^n = a^(m+n)
Dividing exponents with the same base: a^m / a^n = a^(m-n)

Q: Can I use a calculator to multiply and simplify rational expressions?

A: Yes, you can use a calculator to multiply and simplify rational expressions. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

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

A: A rational expression is in its simplest form when there are no common factors between the numerator and denominator.

Q: Can I simplify a rational expression that has a negative exponent?

A: Yes, you can simplify a rational expression that has a negative exponent. However, you need to be careful when canceling common factors, as the negative exponent may not be a common factor.

Conclusion

Multiplying and simplifying rational expressions is a crucial skill in algebra, and with practice, you can master it. Remember to always follow the order of operations, factor the numerator and denominator, cancel common factors, and simplify the resulting expression. By following the steps outlined in this article, you can simplify even the most complex rational expressions.