Problem 21 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{x}{x-2} \cdot \frac{x^{2}-3 x+2}{x-1}$$

Step-by-Step Solution

Answer

The product simplifies to \(x\).

Step 1: Factoring

The first step is to factorize the polynomials in the numerators and denominators if possible. However, in this case, the numerator \(x\) in the first fraction and the denominator \(x-1\) in the second fraction are already in simplest form. The denominator of the first fraction and the numerator of the second may be factored into: \(x^{2}-3 x+2 = (x-1)(x-2)\). Thus the expression becomes \(\frac{x}{x-2} \cdot \frac{(x-1)(x-2)}{x-1}\).

Step 2: Cancel common factors

The next step in simplifying rational functions is to cancel common factors from the numerators and denominators of the fractions. Here, \(x-2\) from the denominator of the first fraction and the numerator of the second fraction can be canceled. Similarly, \(x-1\) from the numerator of the second fraction and the denominator can also be canceled. Thus, the expression simplifies to \(x\).

Step 3: Final Answer

The rational function simplifies to \(x\), which is the final answer.

Key Concepts

Simplifying Expressions

When dealing with rational expressions, the primary goal is to simplify the given expression. Simplifying involves reducing the expression to its simplest form, where no further simplification can occur. In the exercise provided, we are working with the product of two fractions: \( \frac{x}{x-2} \cdot \frac{x^{2}-3x+2}{x-1} \).

• To simplify, first observe if any terms can be reduced through factoring or canceling common factors, as discussed later on.
• After simplification, if no like terms can be further canceled or combined, you have successfully simplified the expression.

Breaking it down into manageable parts often helps in understanding and achieving the task more effectively. It's similar to cleaning your desk: start with one pile and gradually work through the clutter until your workspace is clear.

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. It involves breaking down a polynomial into simpler "factors" that, when multiplied together, return the original polynomial. In the expression \( \frac{x}{x-2} \cdot \frac{x^2-3x+2}{x-1} \), factoring helps us see common factors that can be canceled.Consider the polynomial \( x^2 - 3x + 2 \). This can be factored into \((x-1)(x-2)\). Here's how:

• Identify two numbers that multiply to \( 2 \) (the constant term) and add to \( -3 \) (the coefficient of \( x \)) .
• The numbers -1 and -2 fit these requirements, as \(-1 \times -2 = 2\) and \(-1 + -2 = -3\).
• Thus, \( x^2 - 3x + 2 = (x-1)(x-2) \).

This factorizing reveals hidden simplicity within polynomials and is key to the next step of simplification.

Canceling Common Factors

Once polynomials have been factored, the next step is to cancel common factors from the numerators and denominators of the fractions in a rational expression. This is akin to removing duplicate items when organizing your room - it makes things much cleaner and neater. In our problem, after factoring \( x^2-3x+2 \) to \((x-1)(x-2)\), the expression becomes \( \frac{x}{x-2} \cdot \frac{(x-1)(x-2)}{x-1} \).

• Here, \(x-2\) from the denominator of the first fraction and the numerator of the second fraction cancels out.
• Similarly, \(x-1\) is present in both the numerator of the second fraction and the denominator. These can also be canceled.

Upon canceling these common factors, the remaining expression is simply \(x\). Cancellations must be done carefully to avoid eliminating terms that should actually be in the final answer. By enforcing this, you ensure the expression is neatly in its simplest form.