Problem 19 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{3(x+2)(x-1)}{6(x-1)^{2}} $$

Step-by-Step Solution

Answer

\( \frac{x+2}{2(x-1)} \)

Step 1: Factor the Numerator and the Denominator

In the expression \( \frac{3(x+2)(x-1)}{6(x-1)^2} \), the numerator is already factored as \( 3(x+2)(x-1) \). The denominator can be factored as \( 6(x-1)^2 \).

Step 2: Cancel Common Factors

Observe the common factors in the numerator and the denominator. Both have \((x-1)\), so cancel out one \((x-1)\) from the numerator and one from the denominator, simplifying the expression to \( \frac{3(x+2)}{6(x-1)} \).

Step 3: Simplify the Coefficient Fraction

Now, simplify the coefficients \( \frac{3}{6} \) to \( \frac{1}{2} \). Thus, the expression becomes \( \frac{1(x+2)}{2(x-1)} \), which simplifies further to \( \frac{x+2}{2(x-1)} \).

Key Concepts

Factoring Expressions

Factoring is a crucial step in simplifying rational expressions. It's the process where we break down expressions into simpler terms that, when multiplied, give back the original expression. In our exercise, the expression is already factored:

• The numerator: \(3(x+2)(x-1)\) consists of a constant \(3\) and the factors \((x+2)\) and \((x-1)\).
• The denominator: \(6(x-1)^2\) includes the constant \(6\) and a repeated factor \((x-1)(x-1)\).

By recognizing these factors, we set the stage for simplifying the expression further. Always ensure each expression is fully factored to identify all possible common factors.

Canceling Common Factors

Once you've factored both the numerator and the denominator, look for common factors. These are terms that appear in both the numerator and the denominator. In our example:

• The common factor: \((x-1)\) appears once in the numerator and twice in the denominator.

By canceling out one \((x-1)\) from both the numerator and the denominator, we reduce the expression to a simpler form. It's like reducing the size of a fraction by dividing both its top and bottom by the same number. Be careful not to cancel out non-common terms. This step is vital to ensure you do not alter the value of the expression.

Simplifying Fractions

Simplifying fractions involves reducing the expression to its most simplified form. After canceling common factors, the expression from our example becomes \( \frac{3(x+2)}{6(x-1)} \). Here, we focus on reducing the coefficient terms:

• The coefficient fraction \(\frac{3}{6}\) simplifies to \(\frac{1}{2}\).

Simplifying numerical fractions like this helps in achieving a cleaner expression. Thus, the expression further simplifies to \( \frac{x+2}{2(x-1)} \). When simplifying, always check if there are any other terms - numerical or variable-based - that can be reduced without altering the expression's value.