Hey guys! Ever stumbled upon an algebraic expression and felt a little lost? Don't worry, we've all been there. Let's break down the expression (x² + 4x)(x) step by step, making sure you understand exactly what's going on. We will also look at evaluating (m/n)(x) when x = -3. Buckle up, because we're diving into the world of polynomials!
Unraveling (x² + 4x)(x): A Deep Dive
When dealing with algebraic expressions like (x² + 4x)(x), the key is to understand the distributive property. This fundamental principle allows us to multiply a term outside the parentheses by each term inside the parentheses. Think of it like this: you're sharing the 'x' with both the 'x²' and the '4x'. Let's break it down:
First, focus on the core concept of the distributive property. It states that a(b + c) = ab + ac. In our case, 'a' is 'x', 'b' is 'x²', and 'c' is '4x'. So, we need to multiply 'x' by both 'x²' and '4x'.
Next, let's tackle the first multiplication: x * x². Remember the rules of exponents? When multiplying terms with the same base (in this case, 'x'), you add the exponents. 'x' is the same as 'x¹', so we have x¹ * x². Adding the exponents (1 + 2), we get x³. So, x * x² = x³.
Now, let's move on to the second multiplication: x * 4x. Here, we're multiplying the coefficients (the numbers in front of the 'x') and adding the exponents. The coefficient of the first 'x' is 1 (even though it's not explicitly written). So, we have 1 * 4 = 4. Then, we multiply the 'x' terms: x * x. Again, we add the exponents (1 + 1), giving us x². So, x * 4x = 4x².
Finally, we combine the results of our two multiplications. We found that x * x² = x³ and x * 4x = 4x². Adding these together, we get x³ + 4x². This is the simplified form of the expression (x² + 4x)(x).
Therefore, the expression (x² + 4x)(x) simplifies to x³ + 4x². It's important to note that the other options, 5x² and 4x⁴, are incorrect. 5x² would be the result of adding x² and 4x², not multiplying. And 4x⁴ would result from multiplying 4x by x³, not (x² + 4x) by x. Remember to always apply the distributive property correctly and pay attention to the rules of exponents!
Alternatives Explored: Why 5x² and 4x⁴ Don't Fit
It's crucial to understand why the other options presented, 5x² and 4x⁴, are incorrect. This solidifies your understanding of polynomial operations and prevents common mistakes.
Let's first address 5x². This answer is a common mistake arising from the misconception of adding terms instead of multiplying. If you were to add x² and 4x², you would indeed get 5x². However, our original expression involves multiplication, not addition, outside the parentheses. The distributive property dictates that we multiply 'x' with each term inside the parentheses, not simply combine like terms within the parentheses before multiplying.
Now, let's consider 4x⁴. This answer might seem plausible if you incorrectly apply the rules of exponents or misunderstand the distribution process. You might arrive at this answer if you were to multiply 4x by x³, which isn't what the original expression asks us to do. The key is that we're multiplying 'x' by the entire expression (x² + 4x), not just a part of it in a way that leads to x⁴.
To further clarify, let's visualize the distributive property again. Imagine you have 'x' groups, and each group contains (x² + 4x) items. To find the total number of items, you need to multiply 'x' by each part of the group separately (x² and 4x) and then add the results. This is why we get x³ + 4x² and not 5x² or 4x⁴.
Understanding why the incorrect answers are wrong is just as important as understanding why the correct answer is right. It helps you identify common pitfalls and develop a deeper understanding of algebraic manipulations. So, always take a moment to analyze why certain approaches don't work – it's a valuable learning experience!
Evaluating (m/n)(x) for x = -3: A Step-by-Step Guide
Now, let's shift gears and tackle the second part of the problem: evaluating (m/n)(x) for x = -3. This involves substituting a specific value for a variable and simplifying the expression. The expression (m/n)(x) simply means (m/n) multiplied by 'x'.
The first key step is understanding substitution. This is a fundamental concept in algebra where you replace a variable (like 'x') with a given numerical value. In our case, we're replacing 'x' with -3. So, (m/n)(x) becomes (m/n)(-3).
Next, consider the implications of multiplying by a negative number. Remember that multiplying any expression by -3 means we are taking -3 times that expression. So, (m/n)(-3) is the same as -3 multiplied by (m/n).
Now, let's perform the multiplication. We can write -3 as a fraction, -3/1. So, we have (-3/1) * (m/n). When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). This gives us (-3 * m) / (1 * n), which simplifies to -3m/n.
Therefore, evaluating (m/n)(x) for x = -3 results in -3m/n. The final answer remains in terms of 'm' and 'n' because we don't have specific values for these variables. It's crucial to remember that unless we know the values of 'm' and 'n', we cannot simplify the expression further.
In summary, evaluating an algebraic expression for a specific variable value involves substituting the value, applying the order of operations, and simplifying the result. This skill is crucial for solving equations and understanding the behavior of functions.
The Importance of Order of Operations
In this problem, the order of operations might seem straightforward, but it's a concept that's worth emphasizing. Understanding and correctly applying the order of operations is crucial for accurate mathematical calculations, especially when dealing with more complex expressions.
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Let's break down how it applies in our case, even though it's relatively simple.
In the expression (m/n)(-3), the parentheses indicate multiplication. So, we are multiplying the fraction (m/n) by -3. There are no exponents, addition, or subtraction in this direct calculation. The division within the fraction (m/n) is inherent to the fraction itself, but the primary operation we are performing is multiplication.
The potential for confusion often arises when expressions become more complex, involving multiple operations. For example, if we had an expression like (m/n) * (-3 + 2), we would first perform the operation within the parentheses (-3 + 2 = -1) and then multiply the result by (m/n). Similarly, if there were exponents involved, we would evaluate them before multiplication or division.
In the context of our original expression, (x² + 4x)(x), the order of operations implicitly guides our application of the distributive property. We are essentially performing multiplication after the implicit addition within the parentheses is conceptually addressed through distribution.
So, while the order of operations might seem like a minor point in this specific problem, mastering it is fundamental to your overall mathematical proficiency. Always remember PEMDAS and apply it consistently to avoid errors in your calculations!
Real-World Applications and Why This Matters
You might be wondering,
