Problem 47 - Algebra 2

Question

Simplify each expression. $$ \frac{y+1}{y-1}+\frac{y+2}{y-2}+\frac{y}{y^{2}-3 y+2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{2y^2 + y - 4}{(y-1)(y-2)} \).

Step 1: Find a Common Denominator

To simplify the expression \( \frac{y+1}{y-1}+\frac{y+2}{y-2}+\frac{y}{y^{2}-3 y+2} \), first find a common denominator. We notice that \( y^2 - 3y + 2 \) can be factored as \((y-1)(y-2)\). Therefore, the common denominator is \((y-1)(y-2)\).

Step 2: Rewrite Each Fraction with Common Denominator

Convert each fraction to have the common denominator \((y-1)(y-2)\):- \( \frac{y+1}{y-1} = \frac{(y+1)(y-2)}{(y-1)(y-2)} \)- \( \frac{y+2}{y-2} = \frac{(y+2)(y-1)}{(y-1)(y-2)} \)- The third term is already \( \frac{y}{(y-1)(y-2)} \).

Step 3: Expand and Simplify each Numerator

Expand and simplify the numerators:- \((y+1)(y-2) = y^2 - 2y + y - 2 = y^2 - y - 2\)- \((y+2)(y-1) = y^2 - y + 2y - 2 = y^2 + y - 2\)

Step 4: Combine the Fractions

Combine the fractions over the common denominator:\(\frac{y^2 - y - 2}{(y-1)(y-2)} + \frac{y^2 + y - 2}{(y-1)(y-2)} + \frac{y}{(y-1)(y-2)} = \frac{y^2 - y - 2 + y^2 + y - 2 + y}{(y-1)(y-2)}\)

Step 5: Simplify the Combined Numerator

Simplify the numerator of the combined fraction:- Combine like terms: \( y^2 + y^2 = 2y^2 \), \( -y + y + y = y \), and \( -2 - 2 = -4 \).- The numerator becomes \( 2y^2 + y - 4 \).

Step 6: Present the Simplified Expression

The expression simplifies to:\[\frac{2y^2 + y - 4}{(y-1)(y-2)}\]

Key Concepts

Common Denominator

When working with algebraic fractions, especially when adding or subtracting them, finding a common denominator is essential. In the given problem, each fraction has a different denominator.
To simplify the expression, you first need to determine a common denominator that can accommodate all the different denominators in your equation. This is similar to finding the least common multiple but applied to denominators.

• In our problem, the denominators are \(y-1\), \(y-2\), and \(y^2-3y+2\).
• The expression \(y^2-3y+2\) can be factored into \((y-1)(y-2)\), which helps us identify our common denominator.
• Thus, the common denominator for all terms in the expression is \((y-1)(y-2)\).

Using a common denominator allows us to combine fractions more easily, as it ensures each fraction can be expressed equivalently over the same base.

Factoring

Factoring is a method used to simplify expressions or solve equations by finding expressions that multiply to form the original expression. It involves breaking down a complex expression into simpler parts.
In our exercise, we have a quadratic expression \(y^2 - 3y +2\), which can be simplified through factoring:

• This expression is a simple trinomial that factors into \((y-1)(y-2)\).
• Factoring helps us to simplify rational expressions and find common denominators more easily.

Recognizing factoring opportunities in algebraic expressions is crucial because it reduces complexity and often unveils solutions or simplifications that aren't immediately visible.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms with the same variables are added or subtracted together. This step is necessary for simplifying complex expressions into their simplest forms.
After aligning all terms to have a common denominator, you then focus on combining terms in the numerator:

• Terms like \(y^2\) and \(y^2\) combine to form \(2y^2\).
• Terms like \(-y\), \(y\), and another \(y\) combine to make \(y\).
• Constant terms like \(-2\) and \(-2\) combine to \(-4\).

By gathering like terms, you consolidate the expression, making subsequent operations easier and clearer, revealing a more understandable final form.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Understanding these types of expressions and how to manipulate them is vital in algebra.
In this context, simplifying a rational expression is about consolidating it into its simplest possible form.

• You need to manage operations like finding a common denominator, factoring, and then combining terms — all steps that contribute to this simplification process.
• The rational expression we aimed to simplify is initially given as multiple fractions, each with its denominator.
• Utilizing algebra skills like factoring and combining like terms, we ultimately simplified it to a single fraction: \(\frac{2y^2 + y - 4}{(y-1)(y-2)}\).

Manipulating rational expressions requires strong knowledge in polynomial operations and a systematic approach to solving and simplifying complex expressions.