Problem 16 - Algebra 2 and Trigonometry

Question

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{3-\frac{9}{x}}{x-8+\frac{15}{x}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3}{x - 5}\), undefined for \(x = 0, 3, 5\).

Step 1: Identify Undefined Values

First, find the values of the variable that make the denominators equal to zero, as these will make the expression undefined. For the whole expression, the denominator is \(x - 8 + \frac{15}{x}\). The secondary denominators within the expression are the \(x\) in both \(\frac{9}{x}\) and \(\frac{15}{x}\).Find values for which the denominators are zero:- For \(\frac{9}{x}\), \(x eq 0\).- For \(x - 8 + \frac{15}{x} \), solve \(x - 8 + \frac{15}{x} = 0\): Combine into a single fraction: \(\frac{x^2 - 8x + 15}{x} = 0\) leads to solving the quadratic equation \(x^2 - 8x + 15 = 0\). Factoring gives \((x-3)(x-5) = 0\), thus \(x = 3\) and \(x = 5\).In conclusion, the expression is undefined for \(x = 0\), \(x = 3\), and \(x = 5\).

Step 2: Combine Components into Single Fractions

Rewrite the expression \(\frac{3-\frac{9}{x}}{x-8+\frac{15}{x}}\) with each part having a common denominator.- Numerator: \(3 - \frac{9}{x} = \frac{3x}{x} - \frac{9}{x} = \frac{3x - 9}{x}\).- Denominator: \(x - 8 + \frac{15}{x} = \frac{x^2}{x} - \frac{8x}{x} + \frac{15}{x} = \frac{x^2 - 8x + 15}{x}\).

Step 3: Simplify the Complex Rational Expression

With the numerator and denominator written with like denominators:1. \(\frac{3x - 9}{x}\) is the numerator.2. \(\frac{x^2 - 8x + 15}{x}\) is the denominator.Combine these into a single division by multiplying by the reciprocal: \[ \frac{3x - 9}{x} \div \frac{x^2 - 8x + 15}{x} = \frac{3x - 9}{x} \times \frac{x}{x^2 - 8x + 15} = \frac{3x-9}{x^2 - 8x + 15}\]

Step 4: Further Simplify the Expression

Notice that \(3x - 9\) can be factored further as: \[3x - 9 = 3(x - 3)\]So, the expression becomes:\[\frac{3(x-3)}{(x-3)(x-5)}\]Cancel \(x-3\) from the numerator and denominator (as long as \(x eq 3\), where it is undefined): \[\frac{3}{x - 5}\]

Key Concepts

Undefined Values

In mathematics, especially when dealing with fractions and rational expressions, it's crucial to know when an expression is defined. A fraction is undefined when its denominator equals zero. This is because division by zero is not possible. Let's dive into the steps to find undefined values in our complex rational expression: \[ \frac{3-\frac{9}{x}}{x-8+\frac{15}{x}} \].

• For individual fractions like \( \frac{9}{x} \) and \( \frac{15}{x} \), the expressions become undefined if \( x = 0 \), since the denominator \( x \) cannot be zero.
• The larger denominator \( x - 8 + \frac{15}{x} \) requires setting \( x^2 - 8x + 15 = 0 \), a quadratic equation. Solving this equation gives values \( x = 3 \) and \( x = 5 \), which also make the original expression undefined as the denominator would become zero.

In conclusion, the expression is undefined for \( x = 0 \), \( x = 3 \), and \( x = 5 \). Remember, identifying these values is essential, as they determine where your expression holds meaning.

Factoring Quadratics

Factoring quadratics is a technique used to break down a quadratic equation, like \( x^2 - 8x + 15 \), into simpler components. This is a crucial step in simplifying expressions containing quadratics. Let's see how it works:

• First, notice that a quadratic expression is typically in the form \( ax^2 + bx + c \). Our quadratic, \( x^2 - 8x + 15 \), has \( a = 1 \), \( b = -8 \), and \( c = 15 \).
• The goal is to express this equation as a product of two binomials, i.e., \( (x - p)(x - q) \).
• Identify \( p \) and \( q \) such that \( p+q = -8 \) and \( p \times q = 15 \). Here, the numbers 3 and 5 fit these conditions, so we can factor the quadratic as \( (x-3)(x-5) \).

Factoring simplifies calculations and helps in solving or simplifying expressions further. Identifying the right pair of numbers that complete the factorization is key.

Simplifying Expressions

Simplifying involves reducing expressions to their simplest form, making them easier to work with. In the studied expression, \[ \frac{3x - 9}{x^2 - 8x + 15} \], we begin by factoring both the numerator and the denominator to simplify them:

• The numerator \( 3x - 9 \) can be factored into \( 3(x - 3) \).
• The denominator \( x^2 - 8x + 15 \), as previously discussed, factors into \( (x-3)(x-5) \).

Now, let's simplify by canceling the common term \( (x-3) \) from both the numerator and the denominator, transforming our expression into:\[ \frac{3}{x-5} \]This simplification is valid as long as \( x eq 3 \), because division by zero is invalid. Simplifying expressions helps in reducing complexity and makes calculations more straightforward.

Algebraic Fractions

An algebraic fraction is a fraction where the numerator and/or the denominator contain algebraic expressions. Complex rational expressions, like the one in our exercise, are examples of algebraic fractions. Let's break down some key points:

• They behave like regular fractions, meaning they follow the same rules when it comes to operations like addition, subtraction, multiplication, and division.
• To simplify or operate with an algebraic fraction, ensure the denominators are the same, or factor to find common terms that can be canceled.

In our exercise, we had to rewrite the complex expression by finding a common denominator before simplifying. It's akin to finding a common denominator with regular fractions but involves algebraic terms. Mastery of algebraic fractions involves understanding these concepts and becoming comfortable with the operations involved. They play a fundamental role in making more complex mathematical concepts accessible.