Simplifying Algebraic Expressions A Step-by-Step Guide

In mathematics, simplifying algebraic expressions is a fundamental skill that allows us to rewrite complex expressions in a more manageable and understandable form. This process often involves factoring, canceling common factors, and combining like terms. In this comprehensive guide, we will delve into the process of simplifying the given expression, ${\frac{4x + 36}{x^2 + 9x}}$, providing a step-by-step explanation to ensure clarity and understanding.

Step 1: Factoring the Numerator

The first step in simplifying this algebraic expression is to factor the numerator, which is ${4x + 36}$. Factoring involves identifying common factors within the expression and extracting them. In this case, we can observe that both terms, ${4x}$ and ${36}$, are divisible by 4. Therefore, we can factor out the 4 from the numerator:

${ 4x + 36 = 4(x + 9) }$

By factoring out the 4, we have rewritten the numerator in a more simplified form. This step is crucial as it allows us to identify potential common factors with the denominator, which we will address in the next step. Factoring is a cornerstone of algebraic simplification, and mastering this technique is essential for tackling more complex expressions.

Step 2: Factoring the Denominator

Next, we turn our attention to the denominator, which is ${x^2 + 9x}$. Similar to the numerator, we need to identify common factors within the denominator. In this case, we can observe that both terms, ${x^2}$ and ${9x}$, have a common factor of ${x}$. Factoring out the ${x}$ from the denominator, we get:

${ x^2 + 9x = x(x + 9) }$

This step is analogous to factoring the numerator, and it's equally important. By factoring the denominator, we expose the common factors that can be canceled out, leading to further simplification of the expression. Factoring polynomials often involves recognizing patterns and applying algebraic identities, and this step demonstrates the importance of recognizing the common factor of ${x}$ in the given quadratic expression.

Step 3: Writing the Factored Expression

Now that we have factored both the numerator and the denominator, we can rewrite the original expression with its factored forms:

${ \frac{4x + 36}{x^2 + 9x} = \frac{4(x + 9)}{x(x + 9)} }$

This step is a crucial bridge between the initial expression and its simplified form. By expressing both the numerator and the denominator in their factored forms, we make it easier to identify and cancel out common factors. This is a standard technique in simplifying rational expressions, where factoring plays a pivotal role in revealing the underlying structure of the expression. The rewritten expression clearly shows the common factor of ${(x + 9)}$, which will be the key to the next step.

Step 4: Canceling Common Factors

At this stage, we can observe that both the numerator and the denominator have a common factor of ${(x + 9)}$. In algebraic simplification, we can cancel out common factors that appear in both the numerator and the denominator, provided that the factor is not equal to zero. In this case, we can cancel out the ${(x + 9)}$ factor:

${ \frac{4(x + 9)}{x(x + 9)} = \frac{4}{x} }$

By canceling out the common factor, we have significantly simplified the expression. This step is based on the fundamental principle that dividing both the numerator and the denominator by the same non-zero quantity does not change the value of the fraction. It's important to note that we are implicitly assuming that ${x + 9 \neq 0}$, or ${x \neq -9}$, as division by zero is undefined. This cancellation step is a core technique in simplifying rational expressions and is applicable to a wide range of algebraic problems.

Step 5: Stating the Simplified Expression

After canceling the common factors, we are left with the simplified expression:

${ \frac{4}{x} }$

This is the simplified form of the original expression ${\frac{4x + 36}{x^2 + 9x}}$. By factoring and canceling common factors, we have reduced the expression to its simplest form. This simplified expression is easier to work with in further calculations or analysis. Simplifying expressions is a crucial skill in algebra, as it allows us to manipulate and solve equations more efficiently. The final simplified expression, ${\frac{4}{x}}$, is a testament to the power of factoring and canceling in algebraic simplification.

Final Answer

Therefore, the simplified form of the expression ${\frac{4x + 36}{x^2 + 9x}}$ is:

${ \frac{4}{x} }$

This corresponds to option A, where the simplified answer is ${\frac{4}{x}}$.

In conclusion, simplifying algebraic expressions involves a series of steps, including factoring, identifying common factors, and canceling them out. By following these steps systematically, we can transform complex expressions into simpler, more manageable forms. This skill is essential in algebra and is used extensively in various mathematical applications. Remember to always check for common factors and cancel them out to achieve the most simplified form of the expression.