Problem 61 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3 x}{2 x+5}+1 $$

Step-by-Step Solution

Answer

\( \frac{5(x + 1)}{2x+5} \) in simplest form.

Step 1: Rewrite the Constant as a Rational Expression

Start by rewriting the number 1 as a rational expression with the same denominator as the other term. The expression becomes \( \frac{3x}{2x+5} + \frac{2x+5}{2x+5} \).

Step 2: Combine the Fractions

Now, since both fractions have the same denominator, you can add the numerators directly. The new expression is \( \frac{3x + (2x + 5)}{2x+5} \).

Step 3: Simplify the Expression

Simplify the expression in the numerator. Combine like terms to get \( \frac{5x + 5}{2x+5} \).

Step 4: Factor and Simplify Further if Possible

Factor the numerator if possible. The numerator can be factored as \( 5(x + 1) \), giving \( \frac{5(x + 1)}{2x+5} \). Check if any further simplification is possible, but in this case, it is already in simplest form since there are no common factors to cancel.

Key Concepts

Rational Numbers

Rational numbers are the foundation of rational expressions, which are fractions involving polynomials. These numbers can be expressed as the quotient of two integers, where the denominator is not zero. This means any number which can be written as a fraction

• a/b is a rational number
• a and b are integers
• b ≠ 0

For example, the number 1 can be expressed as 1/1 or even 2/2, making it rational. Understanding rational numbers helps us manipulate rational expressions, like rewriting constants in terms of expressions with common denominators.

Adding Fractions

When dealing with fractions, especially rational expressions, adding them requires a common denominator. This is similar to adding fractions in basic arithmetic. Before you proceed:

• Ensure both fractions have the same denominator.
• If they do not, find an equivalent fraction for one or both fractions.
• Once the denominators are the same, simply add the numerators.

For example, to add \( \frac{3x}{2x+5} \) and 1, we first rewrite 1 as a fraction with the denominator \(2x+5\), resulting in \( \frac{2x+5}{2x+5} \). Then, combine the fractions: \( \frac{3x + (2x+5)}{2x+5} \).

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. In the context of rational expressions, once you've combined the numerators, you should look for opportunities to simplify:

• Combine like terms in the numerator.
• See if further reduction is possible by factoring.
• Ensure the denominator remains unchanged.

For instance, after combining like terms in \( \frac{3x + 2x + 5}{2x+5} \), you get \( \frac{5x + 5}{2x+5} \), which simplifies the expression efficiently.

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying more complex rational expressions. It involves breaking down a polynomial into its smallest components, often referred to as its factors. Here’s how you approach it:

• Look for a common factor in each term.
• Factor out the greatest common factor.
• Rewrite the expression based on these factors.

In the exercise, the numerator \(5x + 5\) can be factored into \(5(x + 1)\). This gives the expression \( \frac{5(x + 1)}{2x+5} \). Factoring helps simplify and sometimes further reduce rational expressions by canceling out common factors if present.