Find The Sum Of The Terms:${ \frac{x}{x^2+3x+2} + \frac{3}{x+1} }$The Numerator Of The Simplified Sum Is { \square$}$.

Mar 13, 2025 by ADMIN 120 views

**Finding the Sum of Rational Expressions: A Step-by-Step Guide**

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Introduction

In algebra, finding the sum of rational expressions is a crucial skill that can be applied to various mathematical problems. Rational expressions are fractions that contain variables in the numerator and/or denominator. In this article, we will explore how to find the sum of two rational expressions, specifically the sum of $\frac{x}{x^2+3x+2}$ and $\frac{3}{x+1}$ .

Understanding Rational Expressions

A rational expression is a fraction that contains variables in the numerator and/or denominator. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when working with rational expressions, we must be careful to simplify the resulting expression to its lowest terms.

Simplifying Rational Expressions

To simplify a rational expression, we need to factor the numerator and denominator, if possible, and then cancel out any common factors. This process is called factoring out the greatest common factor (GCF).

Factoring the Numerator and Denominator

Let's factor the numerator and denominator of the given rational expressions:

$\frac{x}{x^2+3x+2}$ : The numerator is already factored, but the denominator can be factored as $(x+1)(x+2)$ .
$\frac{3}{x+1}$ : The numerator is a constant, and the denominator is already factored as $(x+1)$ .

Finding the Sum of the Rational Expressions

Now that we have factored the numerator and denominator of each rational expression, we can find the sum by combining the two expressions.

Combining the Rational Expressions

To combine the rational expressions, we need to find a common denominator. In this case, the common denominator is $(x+1)(x+2)$ .

from sympy import symbols, simplify
x = symbols('x')

expr1 = x / (x**2 + 3*x + 2)
expr2 = 3 / (x + 1)

common_denominator = (x + 1) * (x + 2)

combined_expr = simplify(expr1 + expr2)

Simplifying the Combined Expression

Now that we have combined the rational expressions, we can simplify the resulting expression to its lowest terms.

Simplifying the Combined Expression

To simplify the combined expression, we can cancel out any common factors between the numerator and denominator.

# Simplify the combined expression
simplified_expr = simplify(combined_expr)

Conclusion

In this article, we have explored how to find the sum of two rational expressions, specifically the sum of $\frac{x}{x^2+3x+2}$ and $\frac{3}{x+1}$ . We have factored the numerator and denominator of each rational expression, combined the expressions, and simplified the resulting expression to its lowest terms. By following these steps, we can find the sum of any two rational expressions.

Final Answer

The final answer is $\boxed{\frac{x+3}{x+2}}$ .

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Introduction

In our previous article, we explored how to find the sum of two rational expressions, specifically the sum of $\frac{x}{x^2+3x+2}$ and $\frac{3}{x+1}$ . In this article, we will address some of the most frequently asked questions related to rational expression summation.

Q&A

Q: What is the difference between adding rational expressions and adding fractions?

A: Adding rational expressions is similar to adding fractions, but with the added complexity of variables in the numerator and/or denominator. When adding fractions, we need to find a common denominator and then add the numerators. When adding rational expressions, we need to factor the numerator and denominator, find a common denominator, and then add the numerators.

Q: How do I know if the rational expressions can be added?

A: Rational expressions can be added if they have a common denominator. If the denominators are different, we need to find a common denominator before adding the expressions.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators of the rational expressions. In the case of the expressions $\frac{x}{x^2+3x+2}$ and $\frac{3}{x+1}$ , the common denominator is $(x+1)(x+2)$ .

Q: How do I find the common denominator?

A: To find the common denominator, we need to factor the denominators of the rational expressions and then find the LCM of the factors.

Q: Can I add rational expressions with different signs?

A: Yes, you can add rational expressions with different signs. When adding rational expressions with different signs, we need to follow the rules of sign addition.

Q: Can I subtract rational expressions?

A: Yes, you can subtract rational expressions. When subtracting rational expressions, we need to follow the rules of sign subtraction.

Q: How do I simplify the sum of rational expressions?

A: To simplify the sum of rational expressions, we need to factor the numerator and denominator, cancel out any common factors, and then simplify the resulting expression.

Q: Can I multiply rational expressions?

A: Yes, you can multiply rational expressions. When multiplying rational expressions, we need to multiply the numerators and denominators separately and then simplify the resulting expression.

Q: Can I divide rational expressions?

A: Yes, you can divide rational expressions. When dividing rational expressions, we need to invert the second expression and then multiply the two expressions.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to rational expression summation. We have covered topics such as adding rational expressions, finding the common denominator, simplifying the sum, and multiplying and dividing rational expressions. By following these guidelines, you can confidently add, subtract, multiply, and divide rational expressions.

Final Tips

Always factor the numerator and denominator of rational expressions before adding or subtracting them.
Find the common denominator before adding or subtracting rational expressions.
Simplify the resulting expression after adding or subtracting rational expressions.
Multiply and divide rational expressions by inverting the second expression and then multiplying or dividing the numerators and denominators.

By following these tips, you can master the art of rational expression summation and become proficient in algebra.