Problem 21 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{3 x y}{9 x y+6 x^{2} y^{3}}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{1}{3 + 2xy^2}\); it is defined for all real \(x\) and \(y\).

Step 1: Factor the Numerator and Denominator

First, identify the common factors in both the numerator and the denominator of the given expression. The numerator is already factored as \(3xy\). In the denominator, \(9xy\) and \(6x^2y^3\) have a common factor. Factoring out \(3xy\), we get the denominator as \(3xy(3 + 2xy^2)\).

Step 2: Simplify the Expression

Now divide both the numerator and the denominator by their greatest common factor, which is \(3xy\), to simplify the expression. So, the expression becomes \(\frac{3xy}{3xy(3 + 2xy^2)} = \frac{1}{3 + 2xy^2}\).

Step 3: Identify the Undefined Values

An expression is undefined when its denominator is zero. Set the denominator equal to zero: \(3 + 2xy^2 = 0\). Solving for \(xy^2\), we get \(xy^2 = -\frac{3}{2}\). However, since \(xy^2\) includes only squares, it cannot be negative. Therefore, no real solution exists for zeros, meaning the denominator is never zero for any real \(x\) and \(y\), and thus there are no undefined values for real numbers.

Key Concepts

Factoring Polynomials

Factoring polynomials is one of the key skills in algebra, especially when dealing with rational expressions. It involves breaking down a polynomial into its simplest building blocks, or factors, similar to breaking down a number into prime factors. For example, consider the expression \(9xy + 6x^2y^3\). We need to identify the greatest common factor (GCF) of the terms present.

• The term \(9xy\) can be factored as \(3 \times 3 \times x \times y\).
• The term \(6x^2y^3\) can be factored as \(2 \times 3 \times x \times x \times y \times y \times y\).

The common factors between these two terms are \(3 \times x \times y\), making \(3xy\) the GCF. Factoring this out from each term, we are left with the expression: \(3xy(3 + 2xy^2)\). This step simplifies polynomial manipulation in any algebraic context, laying the groundwork for simplification.

Undefined Values in Fractions

In algebra, certain values can make a fraction undefined, which typically occurs if the denominator equals zero. For a rational expression, it's crucial to identify these values to ensure the expression is well-defined at all other points.

In this expression, we started with \(\frac{3xy}{9xy + 6x^2y^3}\) and simplified it to \(\frac{1}{3 + 2xy^2}\). The fraction becomes undefined when the denominator \(3 + 2xy^2\) is zero. Setting up the equation:
\[3 + 2xy^2 = 0\].
Solving for \(xy^2\), we subtract 3 from both sides, yielding:
\[2xy^2 = -3\]
\[xy^2 = -\frac{3}{2}\].
Since the square of a number \((xy^2)\) is never negative for real numbers, this equation has no real solutions. Hence, the expression does not have any real undefined values, meaning there are no values for \(x\) or \(y\) in this expression that would make the denominator zero.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and arithmetic operators. Rational expressions, like \(\frac{3xy}{9xy + 6x^2y^3}\), are fractions containing algebraic expressions in their numerator and/or denominator. Simplifying such expressions necessitates understanding each part's role.

• The expression \(3xy\) in the numerator signifies that every term within the expression must consider \(x\) and \(y\) as factors.
• The expression \(9xy + 6x^2y^3\) in the denominator combines terms that further involve variables \(x\) and \(y\), emphasizing the need for factoring.

Simplifying involves discovering the GCF and removing it. After canceling out common terms, what's left is a reduced fraction, \(\frac{1}{3 + 2xy^2}\). This process showcases how algebraic expressions operate within rational contexts, emphasizing simplification's importance to solving complex problems efficiently and clearly.