Problem 10 - Algebra and Trigonometry

Question

simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{x^{2}-8 x+16}{3 x-12} $$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x-4}{3}\). The number that must be excluded from the domain of the simplified rational expression is \(x=4\).

Step 1: Simplify the numerator and the denominator

First, factorize the numerator and the denominator if possible. The numerator \(x^{2}-8x+16\) can be factored into \((x-4)^{2}\), and the denominator \(3x-12\) can be factored into \(3(x-4)\). So the expression can be written as \(\frac{(x-4)^{2}}{3(x-4)}\).

Step 2: Simplify the rational expression

We observe that \(x-4\) is common in both numerator and denominator, cancel out this term to simplify the expression. So the simplified expression is \(\frac{x-4}{3}\). However, this simplification is only valid when \(x \neq 4\), because if \(x=4\), then the original expression would be undefined since division by zero is undefined.

Step 3: Find the excluded values

The only value that is excluded from the domain of the original rational expression is \(x=4\) as it makes the denominator zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is crucial when simplifying rational expressions. To factor a polynomial, look for patterns or common factors that can be taken out to rewrite the expression in a simpler form. An example of a common pattern is a perfect square trinomial, which is in the form of \(a^2 + 2ab + b^2\) and can be factored into \((a + b)^2\). Similarly, the difference of squares is also a frequent pattern given by \(a^2 - b^2\) and factors into \((a + b)(a - b)\).

In the given exercise, the numerator \(x^2 - 8x + 16\) is a perfect square trinomial and can be factored into \((x - 4)^2\), which is observed by recognizing it as \((a - b)^2\) where \(a = x\) and \(b = 4\). Being able to factor polynomials simplifies the rational expression, making it easier to solve and understand the domain of the expression.

Excluded Values in Rational Expressions

When working with rational expressions, excluded values are numbers that make the denominator equal to zero. Identifying these values is an essential step because they indicate the limits of the expression's domain. To find the excluded values, set the denominator equal to zero and solve for the variable.

In the step-by-step solution, the denominator after factoring is \(3(x - 4)\). By setting \(x - 4 = 0\), you find that the excluded value is \(x = 4\) because it would cause division by zero, which is undefined in mathematics. This is an important concept to master as it involves understanding the behavior of rational expressions and ensuring that they are valid over their domains.

Domain of Rational Expressions

The domain of a rational expression comprises all the real numbers for which the expression is defined. It excludes any values that would make the denominator zero. To determine the domain of an expression, after excluding such values, consider all other possible numbers that the variable can take.

For example, the simplified expression \(\frac{x-4}{3}\) is defined for all real numbers except for \(x = 4\), which was identified as an excluded value. The domain of this rational expression is therefore all real numbers except 4, which can be represented as \({x \in \mathbb{R} | x eq 4}\). Recognizing the domain is critical as it highlights the range over which the expression can operate. Without understanding the domain, one can easily make the mistake of using values that invalidate an expression, leading to incorrect conclusions or undefined results.