Problem 29 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{4-2(x-1)}{x^{2}-6 x+9}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{-2}{x - 3}\), undefined for \(x = 3\).

Step 1: Simplify the Numerator

First, simplify the expression in the numerator. Start with the expression \(4 - 2(x - 1)\). Distribute the \(-2\) to both terms inside the parentheses: \(4 - 2x + 2\). Now combine like terms, resulting in the expression \(-2x + 6\).

Step 2: Factor the Denominator

Next, factor the quadratic expression in the denominator \(x^2 - 6x + 9\). Recognize that this is a perfect square trinomial. It can be factored as \((x - 3)(x - 3)\) or \((x - 3)^2\).

Step 3: Write the Simplified Expression

With the numerator simplified to \(-2x + 6\) and the denominator factored as \((x - 3)^2\), the expression is now \(\frac{-2x + 6}{(x - 3)^2}\). Further simplification is possible by factoring the numerator.

Step 4: Factor out the Common Factor in the Numerator

Identify a common factor in the simplified numerator \(-2x + 6\). We can factor out \(-2\), giving us \(-2(x - 3)\). Thus, the expression becomes \(\frac{-2(x - 3)}{(x - 3)^2}\).

Step 5: Cancel out Common Factors

Since \(x - 3\) appears in both the numerator and the denominator, cancel one \(x - 3\) from both. The expression simplifies to \(\frac{-2}{x - 3}\).

Step 6: Identify Undefined Values

Determine where the original denominator is zero, as these values are undefined for the expression. Solve \((x - 3)^2 = 0\), which results in \(x = 3\). Therefore, the expression is undefined for \(x = 3\).

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form by eliminating common factors from the numerator and the denominator. This helps to streamline the expression for easier evaluation and use.

To simplify a rational expression:

• First, look at both the numerator and the denominator to identify any common factors.
• Factorize both parts whenever possible, which often involves breaking down complex expressions into products of simpler terms.
• Cancel out common factors that appear in both the numerator and the denominator.

Consider the example: \[\frac{-2(x - 3)}{(x - 3)^2}\]In this case:- The numerator has a factor of \(-2\)and \((x - 3)\),while the denominator includes \((x - 3)^2\).- By canceling out one\((x - 3)\)from the numerator and denominator, the expression simplifies to: \(\frac{-2}{x - 3}\).This process helps us to make the expression simpler and more workable.

Factoring Quadratics

Factoring quadratics is a vital skill in simplifying rational expressions, particularly when dealing with the denominator. Quadratic expressions often form the denominator in rational expressions, and transforming them into simpler factors is crucial.

Quadratic expressions take the form \(ax^2 + bx + c\).To factor them:

• Look for a perfect square trinomial, which can be written as \((a + b)^2\)or \((a - b)^2\).This means the quadratic is the square of a binomial.
• If faced with non-square trinomials, use methods like the quadratic formula or trial and error to determine factor pairs.

In the example above:- The denominator \(x^2 - 6x + 9\)is factored as \((x - 3)^2\),literally the perfect square of \((x - 3)\).This approach allows for easier simplification and identification of potential undefined values.

Undefined Values in Fractions

Understanding undefined values in fractions is crucial to prevent inaccurate computations. In a rational expression, undefined values occur when the denominator equals zero, since division by zero is mathematically undefined.

To find these values in a given rational expression, set the denominator equal to zero and solve for the variable to identify the restrictions. These values must be excluded from the expression's domain.

In our example, the denominator is \((x - 3)^2\).To find undefined values:

• Set \((x - 3)^2 = 0\).
• Solving this gives \(x = 3\),meaning the expression \(\frac{-2}{x - 3}\)is undefined at \(x = 3\).

This knowledge helps in correctly determining where the expression holds or fails, ensuring accurate analysis and computations.