Problem 56 - Algebra for College Students

Question

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{x^{2}+3 x+4 x+12}{2 x^{2}+6 x-x-3}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x + 4}{2x - 1}\), with \(x \neq -3\).

Step 1: Understand the Problem

The goal is to simplify the given rational expression \( \frac{x^{2}+3x+4x+12}{2x^{2}+6x-x-3} \). To do this, we need to factor both the numerator and the denominator by grouping.

Step 2: Factor the Numerator by Grouping

Take the expression in the numerator: \(x^{2}+3x+4x+12\). Group terms to find a common factor: \((x^2 + 3x) + (4x + 12)\). Factor out the common factors from each group: \(x(x + 3) + 4(x + 3)\). Notice that \((x+3)\) is a common factor, so factor it out: \((x + 3)(x + 4)\).

Step 3: Factor the Denominator by Grouping

Take the expression in the denominator: \(2x^{2}+6x-x-3\). Group terms: \((2x^2 + 6x) + (-x - 3)\). Factor out common factors: \(2x(x + 3) - 1(x + 3)\). Here, \((x+3)\) is common, so factor it out: \((x + 3)(2x - 1)\).

Step 4: Simplify the Expression

Now that both the numerator \((x + 3)(x + 4)\) and the denominator \((x + 3)(2x - 1)\) are factored, cancel out the common factor \((x + 3)\). The simplified expression is \(\frac{x + 4}{2x - 1}\).

Step 5: State the Constraint

Note that \((x + 3)\) was a common factor in both the numerator and the denominator. This means \(x eq -3\) to avoid division by zero.

Key Concepts

Factoring by Grouping

Factoring by grouping is a useful technique for simplifying polynomial expressions. When a polynomial has four terms, grouping can help by dividing the terms into pairs and factoring them separately. Let’s see how this works in practice:

• Start by grouping terms that can easily be factored. In the expression \(x^{2}+3x+4x+12\), group as \((x^2 + 3x)\) and \((4x + 12)\).
• Factor each group individually. Here, \(x\) can be factored from \((x^2 + 3x)\), resulting in \(x(x + 3)\), and \(4\) from \((4x + 12)\), giving \(4(x + 3)\).
• If both groups share a common term, combine them. In this case, both have \((x + 3)\), so factor it out, obtaining \((x + 3)(x + 4)\).

Factoring through grouping simplifies expressions by reducing them to simpler, more manageable parts.

Common Factors

Common factors are elements that are shared by different terms or expressions. Identifying and removing common factors can greatly simplify mathematical expressions.

• Begin by inspecting each term or factor. Look for elements that repeat across various groups.
• In our example, we've identified \((x + 3)\) as a common factor in both the numerator and denominator after grouping.
• Pull out this common factor, as it helps reduce the complexity of the entire expression.

Recognizing common factors not only makes calculations easier but also unveils simpler forms of equations that are easier to interpret.

Numerator and Denominator

In rational expressions, the numerator is the top part, while the denominator is the bottom. Both parts play a crucial role since simplifying a rational expression involves factoring both.

• The numerator here is \(x^{2}+3x+4x+12\). After factoring by grouping, this becomes \((x + 3)(x + 4)\).
• The denominator is \(2x^{2}+6x-x-3\). Similarly, it factors to \((x + 3)(2x - 1)\) using the same technique.

Remember that the relationship between the numerator and denominator can change the expression's behavior, especially when considering what values \(x\) cannot take. In our final simplified form, \((x + 4)/(2x - 1)\), this relationship is more apparent after removing common factors.

Division by Zero

In mathematics, division by zero is undefined, meaning you cannot divide any number by zero. When simplifying rational expressions, it’s vital to identify values of variables that would make the denominator zero.

• For the simplified expression \((x + 4)/(2x - 1)\), set \(2x - 1\) equal to zero to find problematic values. Solving \(2x - 1 = 0\) gives \(x = 0.5\).
• Additionally, since \((x + 3)\) was a factor removed during simplification, ensure \(x eq -3\) to maintain the integrity of the original expression.
• Overall, identifying these constraints prevents undefined situations and maintains the validity of the expression.

Being aware of division by zero is crucial for anyone working with rational expressions, ensuring solutions are mathematically sound within their defined domains.