Problem 18 - 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. \(\frac{1+\frac{1}{y}-\frac{6}{y^{2}}}{1+\frac{11}{y}+\frac{24}{y^{2}}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y - 2}{y + 8}\) with constraints \(y \neq 0, -3, -8\).

Step 1: Identify the Denominators

To find where the expression is undefined, identify the denominators in each fraction. In the original expression: \( y \) and \( y^2 \) are denominators that cause the expression to be undefined when \( y = 0 \). Thus, \( y eq 0 \).

Step 2: Simplify the Numerator

Express the numerator \( 1 + \frac{1}{y} - \frac{6}{y^2} \) as a single fraction. The common denominator for all terms is \( y^2 \). Thus, it becomes: \[ \frac{y^2}{y^2} + \frac{y}{y^2} - \frac{6}{y^2} = \frac{y^2 + y - 6}{y^2} \].

Step 3: Simplify the Denominator

Simplify the denominator \( 1 + \frac{11}{y} + \frac{24}{y^2} \) similarly. Use \( y^2 \) as the common denominator, resulting in: \[ \frac{y^2}{y^2} + \frac{11y}{y^2} + \frac{24}{y^2} = \frac{y^2 + 11y + 24}{y^2} \].

Step 4: Simplify the Entire Fraction

We now have: \[ \frac{\frac{y^2 + y - 6}{y^2}}{\frac{y^2 + 11y + 24}{y^2}} = \frac{y^2 + y - 6}{y^2 + 11y + 24} \] because the \( y^2 \) in the numerator and denominator cancel out.

Step 5: Factor the Quadratics

Factor the quadratic expressions if possible. The numerator is factored as \( (y - 2)(y + 3) \), and the denominator factors to \( (y + 3)(y + 8) \).

Step 6: Simplify the Expression

Cancel the common factor \((y + 3)\) from the numerator and the denominator. The expression simplifies to \(\frac{y - 2}{y + 8}\).

Step 7: Identify Additional Restrictions

From factoring, the denonimator \((y + 8)\) implies \( y eq -8 \), and from \( (y + 3) \), \( y eq -3 \). Thus, the expression is undefined for \( y = 0, -3, -8 \).

Key Concepts

Simplifying Fractions

Simplifying fractions means rewriting them in their most reduced form. In the context of complex rational expressions—a fraction made up of fractions themselves—we simplify by first combining the fractions in the numerator and separately in the denominator.

Here's how to simplify a fraction step-by-step:

• Determine the least common denominator (LCD) for the fractional parts.
• Rewrite each fraction with this common denominator.
• Combine the terms in the numerator and combine the terms in the denominator.
• Simplify the overall expression by canceling out any common factors in the numerator and denominator.

For instance, when simplifying \( \frac{1+\frac{1}{y}-\frac{6}{y^2}}{1+\frac{11}{y}+\frac{24}{y^2}} \), using \( y^2 \) as the common denominator brings uniformity, allowing terms to be easily combined, leading to a more straightforward simplification.

Undefined Expressions

Expressions become undefined when any value results in division by zero. In rational expressions, this often occurs in the denominators. Understanding where a fraction is undefined helps identify the values of variables that make the expression non-computable.

For complex rational expressions:

• Examine each denominator in both the numerator and the denominator of the expression.
• Set these denominators equal to zero and solve for the variable.
• The solutions are the values that make the expression undefined.

In our complex rational expression, \( y^2 \) and \( y \) are denominators, leading the expression to be undefined when \( y = 0 \). Post simplification and factoring, additional restrictions \( y = -3 \) and \( y = -8 \) imply further points where the expression is undefined.

Factorization

Factorization involves breaking down a larger expression into a product of simpler terms or factors. This becomes crucial for simplifying rational expressions since it allows common factors to be canceled out, reducing the expression further.

To factor quadratic expressions:

• Identify expressions in the form of \( ax^2 + bx + c \).
• Find two numbers that multiply to \( ac \) and add to \( b \).
• Rewrite the middle term using these two numbers, then group and factor by grouping.

Within the given problem, the numerators \( y^2 + y - 6 \) and the denominator \( y^2 + 11y + 24 \) are factored to \((y - 2)(y + 3)\) and \((y + 3)(y + 8)\), respectively. This enables simplification by canceling out \((y + 3)\) as a common factor.

Quadratic Expressions

Quadratic expressions, often seen as polynomial expressions of degree two, are fundamental in algebra and rational expressions. They take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

Key characteristics include:

• The graph of a quadratic expression is a parabola.
• Quadratics can always be factored, completing the square, or using the quadratic formula, depending on what the scenario demands.
• The solutions to \( ax^2 + bx + c = 0 \) are the expressions' roots, which help in factorizing.

In context, factorizing quadratic expressions in the problem directly simplifies solving and identifying when expressions are undefined. Recognizing the standard factored forms also aids in quickly canceling terms during simplification.