Problem 22 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{20 x+50}{15 x-30}\)

Step-by-Step Solution

Answer

\(\frac{2(2x + 5)}{3(x - 2)}\) is the simplified expression.

Step 1: Factor the Numerator

Begin by factoring the numerator, which is \(20x + 50\). Identify the greatest common factor (GCF) of the terms 20x and 50, which is 10. Factor out 10: \[20x + 50 = 10(2x + 5)\]

Step 2: Factor the Denominator

Next, factor the denominator, \(15x - 30\). The GCF of the terms 15x and -30 is 15. Factor out 15: \[15x - 30 = 15(x - 2)\]

Step 3: Simplification by Cancellation

The rational expression becomes: \[\frac{10(2x + 5)}{15(x - 2)}\]Cancel the common factor of 5 in both the numerator and the denominator:\[\frac{2(2x + 5)}{3(x - 2)}\]

Step 4: Simplified Expression

The simplified rational expression is: \[\frac{2(2x + 5)}{3(x - 2)}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is a foundational skill in simplifying rational expressions. When you have an algebraic expression, such as a polynomial, factoring means writing it as a product of its simpler components. These components are called factors. For instance, if you have the polynomial \( 20x + 50 \), it can be factored by finding common factors of its terms.

To begin factoring, look for the Greatest Common Factor (GCF) among the terms, which simplifies the process. Once you identify the GCF, you can factor it out of each term to write the polynomial as a product of its factors. This step is crucial before cancelling terms in a rational expression. Polynomials may also be factored using other techniques like grouping, or special patterns such as the difference of squares and perfect square trinomials. Understanding these techniques enhances your ability to simplify expressions efficiently.

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest number that divides all the terms in a polynomial without leaving a remainder. Identifying the GCF is the first step in simplifying expressions because it helps reduce the terms to their simplest form.

In our example, to factor \( 20x + 50 \), you start by identifying the GCF of these terms, which is 10. This is because both 20 and 50 can be divided evenly by 10. Once the GCF is found, you divide each term of the polynomial by it and write the expression as \( 10(2x + 5) \). Similarly, for \( 15x - 30 \), the GCF is 15, leading to \( 15(x - 2) \).

Recognizing the GCF not only aids in simplifying expressions but also helps when adding, subtracting, or comparing rational expressions. It's a basic yet powerful tool in algebra.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions or polynomials. These can often seem intimidating, but they follow the same rules as regular fractions. Simplifying them requires factoring and then canceling out common factors, just like in numerical fractions.

For example, the given rational expression \( \frac{20x + 50}{15x - 30} \) involves polynomials on both top and bottom. After factoring the numerator and denominator, you can simplify by canceling common factors. Just remember that whatever operation you perform, it must be valid for the entire expression to keep equivalent relationships.

Cancellation in Algebra

Cancellation in algebra refers to the process of removing common factors in the numerator and denominator of a fraction. This reduces the fraction to its simplest form. It’s similar to canceling out numerical fractions where the numerator and denominator share a common factor.

In simplifying our expression \( \frac{10(2x + 5)}{15(x - 2)} \), we notice that 10 and 15 have a common factor of 5, which can be cancelled. This step simplifies our expression to \( \frac{2(2x + 5)}{3(x - 2)} \). The idea is to "divide out" the common factor, thereby simplifying the expression while maintaining its value.

• Always ensure to simplify both numerator and denominator by their greatest common factors.
• Check the remaining terms to confirm no further simplification is possible.

Cancellation is a handy skill that makes working with algebraic fractions much easier, but should be performed carefully to avoid mistakes.