Problem 6 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{16 x^{3} y^{2}}{-28 x^{2} y}$$

Step-by-Step Solution

Answer

\(\frac{4xy}{-7}\)

Step 1: Identify the Greatest Common Factor (GCF)

Determine the greatest common factor of the numerator and the denominator. For the coefficients 16 and -28, the GCF is 4. For the variables in the numerator \(x^3y^2\) and the denominator \(x^2y\), the GCF is \(x^2y\).

Step 2: Simplify the Coefficient

Divide both the coefficient of the numerator and denominator by their GCF. This gives: \(\frac{16}{4} = 4\) and \(\frac{-28}{4} = -7\).

Step 3: Simplify the Variables

Divide the variables in the numerator and the denominator by the GCF found in Step 1. We have \(\frac{x^3}{x^2} = x\) and \(\frac{y^2}{y} = y\).

Step 4: Write the Simplified Expression

Combine the simplified coefficients and variables to form the simplified fraction: \(\frac{4xy}{-7}\).

Key Concepts

Greatest Common Factor

When simplifying algebraic fractions, identifying the greatest common factor (GCF) is an essential first step. The GCF is the highest number that divides exactly into two or more numbers. For algebraic terms, it is the largest expression that is a factor of each term.

To find the GCF of the coefficients in our example, we look at the numbers 16 and -28. The GCF of these numbers is 4, since 4 is the largest number that divides both 16 and -28 without leaving a remainder.

• For coefficients, list the factors of each number.
• Identify the largest factor common to each list.

When it comes to variables, the process involves identifying the smallest power of each common variable.

• In this exercise, we observe that for the variables in the numerator \(x^3y^2\) and in the denominator \(x^2y\), \(x^2\) is the smallest power of \(x\) common to both, and \(y\) is the smallest for the variable \(y\).

Understanding how to determine the GCF helps in reducing the fraction efficiently.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they contain algebraic expressions in the numerator and/or the denominator. Simplifying these fractions makes them easier to work with and interpret.

An algebraic fraction involves dividing polynomials or monomials. In our example, \(\frac{16x^3y^2}{-28x^2y}\), both the numerator and the denominator are algebraic expressions composed of coefficients and variables.

• The numerator here is \(16x^3y^2\).
• The denominator here is \(-28x^2y\).

Simplification of such fractions is done by following a systematic path:

• Identify and factor out the GCF from both the numerator and the denominator.
• Simplify the fraction by dividing both components by this GCF.

By reducing the algebraic fraction, you make the expression simpler and more concise, which can help in further calculations or evaluations.

Coefficient Simplification

In algebraic fractions, after determining the greatest common factor, the next step is simplifying the coefficients. Simplifying coefficients means dividing the numbers in both the numerator and denominator by their GCF.

Taking our example fraction, \(\frac{16x^3y^2}{-28x^2y}\):

• The coefficients are 16 in the numerator and -28 in the denominator.
• We divide both by their GCF, which is 4.

This operation simplifies \(\frac{16}{4} = 4\) and \(\frac{-28}{4} = -7\).

• Always simplify numeric coefficients first, as it can make handling the variables more straightforward.

After simplifying the coefficients, the operation might change the fraction's sign depending on the numbers involved. Notice here our simplified coefficient is \(\frac{4}{-7}\).

Handling coefficients correctly is crucial for achieving the complete simplification of algebraic fractions.