Problem 94 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{3}{x^{2}-2 x+1}+\frac{1}{x^{2}-3 x+2} $$

Step-by-Step Solution

Answer

\(\frac{4x - 7}{(x-1)^2 (x-2)}\) is the simplified expression.

Step 1: Factor Denominators

First, factor the quadratic expressions in the denominators. For the first expression, the denominator is \(x^{2} - 2x + 1\), which factors to \((x-1)^2\).For the second expression, the denominator is \(x^{2} - 3x + 2\), which factors to \((x-1)(x-2)\).

Step 2: Determine a Common Denominator

The common denominator for the two fractions is the least common multiple of the factored denominators: \((x-1)^2 \, and \, (x-1)(x-2)\). The common denominator is \((x-1)^2(x-2)\).

Step 3: Adjust Fractions to Common Denominator

Transform each fraction so they have the same denominator of \((x-1)^2(x-2)\).For \(\frac{3}{(x-1)^2}\), multiply both the numerator and denominator by \((x-2)\) to get:\(\frac{3(x-2)}{(x-1)^2(x-2)}\).For \(\frac{1}{(x-1)(x-2)}\), multiply both the numerator and denominator by \((x-1)\) to get:\(\frac{1(x-1)}{(x-1)^2(x-2)}\).

Step 4: Combine Fractions

Add the two fractions now that they have a common denominator:\(\frac{3(x-2) + (x-1)}{(x-1)^2(x-2)}\).Simplify the numerator:\(3(x-2) = 3x - 6\) and \(1(x-1) = x-1\).Combine: \(3x - 6 + x - 1 = 4x - 7\).

Step 5: Simplify Final Expression

The combined fraction is:\(\frac{4x - 7}{(x-1)^2(x-2)}\).There are no further common factors that can be cancelled, so this is the simplified expression.

Key Concepts

Factoring Quadratic Expressions

Rational expressions often contain polynomial denominators that require simplification through factoring. A crucial step in simplifying them involves breaking down quadratic expressions into their factors. Consider the expression \(x^2 - 2x + 1\). By identifying common patterns, such as perfect squares or other factorable forms, we determine it simplifies to \((x-1)^2\). Factoring allows us to express the quadratic in terms of simpler binomial expressions. Similarly, for \(x^2 - 3x + 2\), recognizing it as a product of two binomials, it factors to \((x-1)(x-2)\). These factored forms will make it easier to work with the expressions in later steps. Look for familiar patterns such as:

• Perfect squares, like \(a^2 + 2ab + b^2 = (a+b)^2\).
• Difference of squares, where \(a^2 - b^2 = (a-b)(a+b)\).
• Common factor extraction, which involves taking out common factors from all terms.

Breaking down these quadratics sets the stage for finding common denominators and combining fractions.

Common Denominator

When adding or subtracting rational expressions, we need a shared foundation; this is where the common denominator comes into play. Once the denominators are factored, as we did in the first section, we can identify their least common multiple (LCM). This shared denominator harmonizes the fractions for addition.\To determine the common denominator for \((x-1)^2\) and \((x-1)(x-2)\), consider each unique factor. The LCM includes each factor raised to its highest power across the terms. So, the LCM includes \((x-1)^2\) (from the first fraction) and \((x-2)\). Therefore, the common denominator becomes \((x-1)^2(x-2)\).

• Ensure the expression contains all unique factors appearing in any of the original denominators.
• Use the highest count of each factor found to cover both fractions.

Finding the common denominator allows us to rewrite each fraction to perform addition seamlessly by transforming them into equivalently valued terms with the same base.

Combining Fractions

After achieving a common denominator, the task is to adjust and combine the fractions. Each fraction's numerator is modified by multiplying it with any missing terms of the common denominator. This process aligns both numerators over the unified base. \br For \(\frac{3}{(x-1)^2}\), multiplying by \(x-2\) yields \(\frac{3(x-2)}{(x-1)^2(x-2)}\). Likewise, \(\frac{1}{(x-1)(x-2)}\) becomes \(\frac{1(x-1)}{(x-1)^2(x-2)}\) when multiplied by \((x-1)\). \br Adding these, we write:\[\frac{3(x-2) + (x-1)}{(x-1)^2(x-2)}\]Simplify by distributing in the numerators: \(3(x-2) = 3x - 6\); \(1(x-1) = x - 1\). Combine like terms in the numerator to get \(4x - 7\).

• Combine terms carefully, ensuring all distributive steps are clear.
• Simplify wherever possible, focusing on both multiplication and addition/subtraction.

This step ensures the expression reaches its simplest, most effective form, \(\frac{4x - 7}{(x-1)^2(x-2)}\), concluding the simplification of the original rational expressions.