Problem 20 - Algebra 1

Question

Simplify the expression. $$\frac{-3}{x-4} \cdot \frac{x-4}{12(x-7)}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-1}{4(x-7)}\).

Step 1: Identify common factors

In the fractions, observe the factor \(x-4\) that appears in both numerator and denominator. This is a common factor that can simplify the expression.

Step 2: Cancel the common factors

Cancel the common factor \(x-4\) from the numerator and the denominator. Now the new expression will be \(\frac{-3}{12(x-7)}\).

Step 3: Simplify the expression further

In the denominator 12 can be simplified as \(3\times4\). Then, one \(3\) in the numerator and another \(3\) in the denominator cancel each other, resulting in \(\frac{-1}{4(x-7)}\).

Key Concepts

Canceling Common Factors

When simplifying algebraic expressions, canceling common factors is a pivotal step. It helps reduce expressions to their simplest form by eliminating identical terms from the numerator and the denominator. Here's how it works:

• First, identify common factors that appear both in the numerator and the denominator.
• Once identified, these common factors can be eliminated, as dividing by the same non-zero term results in 1, essentially removing them from the expression.

In the example exercise, the common factor is \( x-4 \). By removing this identical term from both parts of the expression, we significantly simplify our work.

This process is essential because it not only reduces the complexity of the expression but also prepares it for further simplification. Always remember to only cancel terms that multiply with the rest of the expression, not terms that are added or subtracted.

Factors

Factors play a critical role in simplifying expressions. A factor is a number or expression that evenly divides another expression without leaving a remainder.

• Every term in a product is a factor.
• In algebraic expressions, factors can be constants, variables, or even other expressions.

In the given problem, identify 12 as a factor of the denominator of the resulting expression \( 12(x-7) \). This can be further broken down into products of its factors: \( 3 \times 4 \times (x-7) \).

Knowing how to identify and work with factors allows us to manipulate expressions effectively. This breaks down the complexity and enables the cancellation of common factors more easily. Always pay attention to identifying and writing down all possible factors, as these are the keys to making simplification possible.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying them involves handling both numerical and algebraic fractions. Here’s how to understand them:

• The simplest form of a rational expression occurs when there are no common factors other than 1 between the numerator and the denominator.
• Skillful manipulation of these expressions often requires identifying and canceling common factors, simplifying both numerical coefficients, and algebraic parts.

In the example provided, you start with a rational expression \( \frac{-3}{x-4} \cdot \frac{x-4}{12(x-7)} \). By canceling the common terms and further reducing numbers through factorization, you simplify it to \( \frac{-1}{4(x-7)} \).

Mastering rational expressions demands practice in factorization and comfort with algebraic manipulation. Confidence with these concepts helps transform seemingly complex expressions into their simplest forms, providing clarity and efficiency in problem-solving.