Summing Rational Expressions Step-by-Step Guide With Examples

In the realm of mathematics, particularly in algebra, finding the sum of rational expressions is a fundamental skill. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, often appear in various mathematical contexts. Mastering the art of summing these expressions is crucial for simplifying complex equations, solving problems in calculus, and tackling real-world applications. In this comprehensive guide, we will delve into the intricacies of summing rational expressions, providing a step-by-step approach that will empower you to conquer these mathematical challenges with confidence.

The expression we will be working with is:

$\frac{x}{x^2+3 x+2}+\frac{3}{x+1}$

Our goal is to simplify this expression into a single rational expression and determine the numerator of the simplified sum. Let's embark on this mathematical journey together, unraveling the steps involved in summing rational expressions.

Step 1: Factor the Denominators

The first crucial step in adding rational expressions is to factor the denominators completely. This will allow us to identify the least common denominator (LCD), which is essential for combining the fractions. In our given expression, we have two denominators: $x^2 + 3x + 2$ and $x + 1$. Let's factor each of them.

The first denominator, $x^2 + 3x + 2$, is a quadratic expression. To factor it, we need to find two numbers that add up to 3 (the coefficient of the x term) and multiply to 2 (the constant term). The numbers 1 and 2 satisfy these conditions. Therefore, we can factor the quadratic as follows:

x^2 + 3x + 2 = (x + 1)(x + 2)

The second denominator, $x + 1$, is already in its simplest form and cannot be factored further.

Now that we have factored the denominators, our expression looks like this:

$\frac{x}{(x + 1)(x + 2)} + \frac{3}{x + 1}$

Step 2: Determine the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that is divisible by all the denominators in the given expression. To find the LCD, we need to consider all the factors present in the denominators, including their highest powers.

In our case, the denominators are $(x + 1)(x + 2)$ and $(x + 1)$. The factors involved are $(x + 1)$ and $(x + 2)$. The highest power of $(x + 1)$ that appears in any denominator is 1, and the highest power of $(x + 2)$ is also 1. Therefore, the LCD is the product of these factors raised to their highest powers:

LCD = (x + 1)(x + 2)

Step 3: Rewrite Each Fraction with the LCD

Now that we have the LCD, we need to rewrite each fraction in the expression so that it has the LCD as its denominator. To do this, we multiply the numerator and denominator of each fraction by the factors that are missing from its denominator compared to the LCD.

For the first fraction, $\frac{x}{(x + 1)(x + 2)}$, the denominator already matches the LCD, so we don't need to do anything to this fraction.

For the second fraction, $\frac{3}{x + 1}$, the denominator is missing the factor $(x + 2)$ compared to the LCD. Therefore, we multiply the numerator and denominator of this fraction by $(x + 2)$:

\frac{3}{x + 1} * \frac{x + 2}{x + 2} = \frac{3(x + 2)}{(x + 1)(x + 2)}

Now, our expression looks like this:

$\frac{x}{(x + 1)(x + 2)} + \frac{3(x + 2)}{(x + 1)(x + 2)}$

Step 4: Add the Numerators

With both fractions now having the same denominator (the LCD), we can add them by adding their numerators and keeping the common denominator. This is a fundamental step in summing rational expressions, as it allows us to combine the fractions into a single expression.

Adding the numerators, we get:

x + 3(x + 2) = x + 3x + 6 = 4x + 6

Therefore, the sum of the two fractions is:

$\frac{4x + 6}{(x + 1)(x + 2)}$

Step 5: Simplify the Result (if possible)

The final step in adding rational expressions is to simplify the resulting fraction, if possible. This involves factoring both the numerator and the denominator and canceling out any common factors. By simplifying the expression, we can present the result in its most concise form, making it easier to work with in subsequent calculations.

In our case, the numerator is $4x + 6$. We can factor out a 2 from this expression:

4x + 6 = 2(2x + 3)

The denominator is $(x + 1)(x + 2)$, which is already in factored form.

Now, our expression looks like this:

$\frac{2(2x + 3)}{(x + 1)(x + 2)}$

There are no common factors between the numerator and the denominator, so we cannot simplify the expression further.

Therefore, the simplified sum of the given rational expressions is:

$\frac{2(2x + 3)}{(x + 1)(x + 2)}$

Step 6: Identify the Numerator

The question asks for the numerator of the simplified sum. From our result, the numerator is:

2(2x + 3) = 4x + 6

Conclusion

In conclusion, the process of summing rational expressions involves several key steps: factoring the denominators, finding the least common denominator (LCD), rewriting each fraction with the LCD, adding the numerators, and simplifying the result. By mastering these steps, you can confidently tackle a wide range of mathematical problems involving rational expressions. The numerator of the simplified sum of the given expression is $4x + 6$. This comprehensive guide has equipped you with the knowledge and skills necessary to unravel the secrets of summing rational expressions. Practice these steps diligently, and you'll find yourself navigating the world of algebra with greater ease and proficiency.

Find the sum of the terms and determine the numerator of the simplified sum: $\frac{x}{x^2+3 x+2}+\frac{3}{x+1}$.

Summing Rational Expressions Step-by-Step Guide with Examples