Simplify (x^3 + 11x^2) / (x^2 + 4x - 77)
Let's break down how to simplify the expression (x^3 + 11x^2) / (x^2 + 4x - 77). Algebraic simplification involves factoring and canceling common terms. This makes complex expressions easier to understand and work with. When we simplify, we aim to represent the expression in its most basic form without changing its value. We will start by factoring both the numerator and the denominator, then see if any common factors can be canceled out. Factoring is a crucial step because it allows us to identify common elements that can be simplified. Once factored, we can easily spot and remove those common factors, leading to a cleaner and more manageable expression. This process not only simplifies the algebra but also provides insights into the structure of the expression, which can be valuable for further analysis or manipulation.
Factoring the Numerator
In simplifying algebraic expressions, the first step is often to factor each part of the expression. Focusing on the numerator, x^3 + 11x^2, we look for common factors. In this case, both terms have 'x' in them. The highest power of 'x' that divides both terms is x^2. So, we factor out x^2 from the numerator:
x^3 + 11x^2 = x^2(x + 11)
Factoring out x^2 simplifies the numerator into a product of x^2 and (x + 11). This factorization is essential because it reveals the underlying structure of the expression, making it easier to identify potential cancellations with factors in the denominator. This is a standard technique in algebra, especially when dealing with rational expressions (expressions that are fractions with polynomials). By factoring, we transform the expression from a sum of terms into a product, which is often more manageable and allows for simplification through cancellation of common factors. This initial factoring step sets the stage for the subsequent steps in simplifying the entire expression.
Factoring the Denominator
Next, we tackle the denominator, which is a quadratic expression: x^2 + 4x - 77. To factor this, we need to find two numbers that multiply to -77 and add up to 4. Let's think about the factors of 77:
- 1 and 77
- 7 and 11
Since we need the numbers to add up to 4, we can use 11 and -7. So, we can rewrite the quadratic expression as:
x^2 + 4x - 77 = (x + 11)(x - 7)
So, the denominator factors into (x + 11)(x - 7). The process of factoring the denominator is crucial because it allows us to identify potential common factors with the numerator. In this case, we've successfully expressed the quadratic expression as a product of two binomials. This factorization is essential for simplifying the overall expression, as it enables us to look for terms that can be canceled out. The ability to quickly and accurately factor quadratic expressions is a fundamental skill in algebra, and it plays a significant role in simplifying more complex expressions.
Simplifying the Expression
Now that we've factored both the numerator and the denominator, we can rewrite the original expression:
(x^3 + 11x^2) / (x^2 + 4x - 77) = [x^2(x + 11)] / [(x + 11)(x - 7)]
Notice that (x + 11) appears in both the numerator and the denominator. We can cancel out this common factor:
[x^2(x + 11)] / [(x + 11)(x - 7)] = x^2 / (x - 7)
Therefore, the simplified expression is x^2 / (x - 7).
Simplifying the expression involves canceling out common factors between the numerator and denominator. This step is valid as long as x ≠-11, because division by zero is undefined. After canceling the common factor (x + 11), we are left with x^2 / (x - 7). This is the simplified form of the original expression. This process highlights the importance of factoring in simplifying algebraic expressions. By factoring both the numerator and denominator, we were able to identify and cancel out a common factor, resulting in a much simpler expression. This not only makes the expression easier to understand and work with but also preserves its value for all valid values of x.
Final Answer
So, to wrap things up, the simplified form of the expression (x^3 + 11x^2) / (x^2 + 4x - 77) is:
x^2 / (x - 7)
And that's it! We've successfully simplified the expression by factoring and canceling common terms. Remember, simplification makes expressions easier to work with while maintaining their original value. Throughout this simplification process, we applied several fundamental algebraic techniques. Factoring allowed us to rewrite the numerator and denominator as products of simpler expressions. Identifying and canceling common factors enabled us to reduce the expression to its simplest form. The final simplified expression, x^2 / (x - 7), is much easier to understand and manipulate than the original expression. This simplification not only makes the expression more manageable but also provides insights into its structure and behavior. By mastering these techniques, you can confidently tackle more complex algebraic expressions and simplify them to their most basic forms.
Simplifying algebraic expressions like (x^3 + 11x^2) / (x^2 + 4x - 77) might seem complex initially, but by systematically breaking it down into smaller, manageable steps, the process becomes quite straightforward. The key steps include factoring the numerator and the denominator, identifying common factors, and then canceling those common factors to arrive at the simplified expression. This approach not only simplifies the algebra but also enhances one's understanding of algebraic manipulation. The ability to simplify expressions is a fundamental skill in algebra and is essential for solving more complex problems in mathematics and related fields. By practicing these techniques, you can become proficient at simplifying various algebraic expressions and gain a deeper appreciation for the elegance and power of algebra.