Problem 38 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{x-3}{x^{2}-6 x+9} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{x-3}\).

Step 1: Analyze the Denominator

The denominator of the expression is \(x^2 - 6x + 9\). This can be identified as a perfect square trinomial. A perfect square trinomial takes the form of \((a-b)^2 = a^2 - 2ab + b^2\). Here, \((x)^2 - 2\cdot3\cdot x + 3^2\), which simplifies to \((x-3)^2\). Thus, the denominator simplifies to \((x-3)^2\).

Step 2: Rewrite the Expression

Using the simplified form of the denominator, \(\frac{x-3}{x^2-6x+9}\) becomes \(\frac{x-3}{(x-3)^2}\).

Step 3: Simplify the Fraction

The numerator \(x-3\) and denominator \((x-3)^2\) share a common factor of \((x-3)\). You can cancel \((x-3)\) from both the numerator and the denominator.After canceling, the fraction simplifies to \(\frac{1}{x-3}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial that can be expressed as the square of a binomial. This simplifies recognizing and simplifying expressions significantly. For example, a generic form of a perfect square trinomial is given by:

• \( (a-b)^2 \) which expands to \( a^2 - 2ab + b^2 \)

To identify a perfect square trinomial, look for terms that match this pattern. Consider the expression \(x^2 - 6x + 9\). Here, \(a = x\) and \(b = 3\). The expanded form, \(x^2 - 2 \times 3 \times x + 3^2\), confirms it fits as a perfect square, simplified as \((x-3)^2\). Recognizing these patterns is the first step in simplifying such expressions, leading to more efficient problem-solving.

Numerator and Denominator

Understanding the terms 'numerator' and 'denominator' is crucial when working with fractions. Simply said, the numerator is the top part of a fraction, and the denominator is the bottom part. They tell us how many parts of a whole we have.For the expression \(\frac{x-3}{x^2-6x+9}\):

• The numerator is \(x-3\). It indicates the part of the whole or the value being divided.
• The denominator is \(x^2 - 6x + 9\), which can further be broken down to \((x-3)^2\).

Grasping this structure helps immensely when simplifying fractions because it tells us what can be manipulated or canceled to make the fraction simpler. Together, they frame the base of how rational expressions are approached.

Cancelling Common Factors

Cancelling common factors in a fraction is a fundamental part of simplifying it. Identifying these factors means both the numerator and the denominator can be divided equally by these terms.For instance, in the expression \(\frac{x-3}{(x-3)^2}\), the common factor is \(x-3\). You can remove this factor from both the numerator and the denominator:

• This operation simplifies the expression:
• Cancel \(x-3\) from both parts, leaving \(\frac{1}{x-3}\).

It is important to ensure that we do not cancel across terms that result in zero denominators, which makes the expression undefined. Ensuring that nothing except the intended common factors is canceled is key to maintaining correctness during this process.

Fraction Simplification

Simplifying fractions is about making them easier to work with by reducing them to their simplest forms. This means getting rid of redundancy or complexity in terms, which can usually be seen by removing common factors.Here’s the general approach:

• Start by identifying any perfect square trinomials or common factors as seen in rational expressions like \(\frac{x-3}{x^2-6x+9}\).
• Rewrite the expression if needed with the discovered simplification, such as \((x-3)^2\) for the denominator.
• Cancel any common factors between the numerator and denominator.

Through these steps, the fraction becomes much more manageable, as seen in transforming \(\frac{x-3}{(x-3)^2}\) to \(\frac{1}{x-3}\). Mastering these techniques enables you to deal with complex algebraic expressions with greater ease and confidence.