Problem 7 - Algebra 1

Question

Which of the following is the simplified form of \(\frac{6+2 x}{x^{2}+5 x+6} ?\) $$\text { A. } \frac{2 x}{x^{2}+5 x}$$ $$\text { B. } \frac{2}{x+5}$$ $$\text { C. } \frac{2}{x+2}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{2}{x+2}\), which matches to option C.

Step 1: Factorization

Factorize the denominator of given expression: \[x^{2}+5 x+6\] which can be factored down to \[(x + 2)(x + 3)\]

Step 2: Simplification

Now, replace the denominator in the given expression \[\frac{6+2 x}{x^{2}+5 x+6}\] with factored form, then it becomes \[\frac{6+ 2x}{(x+2)(x+3)}\]. Divide the numerator and the denominator by 2. It simplifies to \[\frac{3+x}{(x+2)(x+3)}\] or \[\frac{x+3}{(x+2)(x+3)}\]. The (x+3) from numerator and denominator will cancel out.

Step 3: Compare with the given answers

After simplification, it is left with \[\frac{1}{x+2}\] which is equivalent to option C.

Key Concepts

Factorization

Understanding the concept of factorization is crucial in algebra, especially when dealing with rational expressions. Factorization refers to the process of breaking down a complicated expression into simpler, multiply-linked factors. It's like finding the original pieces of a puzzle which, when put together, form the complete picture. In the exercise, the quadratic expression in the denominator \( x^2 + 5x + 6 \) asks us to find such pieces or factors that multiply to give the original expression.

The factors of \( x^2 + 5x + 6 \) are \( x + 2 \) and \( x + 3 \) because these two binomials, when multiplied, will give us the original quadratic expression. That's because the product of the first terms (\(x\) times \(x\)), the outer terms (\(x\) times 3), the inner terms (2 times \(x\)), and the last terms (2 times 3) add up to the original expression. This step is foundational since correctly factorizing leads to the ability to cancel common factors, thereby simplifying the expression efficiently.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In simpler terms, they are like fractions, but instead of numbers, you have expressions involving variables like \(x\). The main goal when working with rational expressions is to simplify them, and the process is similar to simplifying numeric fractions: we search for common factors in both the numerator and the denominator that can be canceled.

In this exercise, the rational expression \(\frac{6+2x}{x^2+5x+6}\) is simplified by factorizing the denominator and maintaining the balance with the numerator. Ensuring the expression remains equivalent throughout the simplification process requires us not to omit any factors unless they are common and can be canceled, and never to cancel terms that are added or subtracted, only those that are multiplied.

Simplifying Algebraic Fractions

Simplifying algebraic fractions, also known as reducing algebraic fractions, is the process of making the fraction as simple as possible. This involves factoring both the numerator and the denominator, then reducing them by canceling out common factors. The end goal is to have the simplest form where the numerator and denominator have no common factors other than 1.

For the given exercise, once the denominator \( x^2 + 5x + 6 \) has been factorized to \( (x+2)(x+3) \) and the expression rewritten as \(\frac{6+2x}{(x+2)(x+3)}\), we notice \(2\) can be factored out from the numerator, resulting in \(\frac{2(3+x)}{(x+2)(x+3)}\). Upon simplifying, the \( (x+3) \) term cancels out from both, leaving us with \(\frac{1}{x+2}\), which is the most reduced form. In doing so, we've applied algebraic principles to transform a potentially complex rational expression into an easier and more understandable form, as seen with Option C.