Factoring The Expression -4.5n + 3

When we talk about factoring an algebraic expression, we're essentially doing the reverse of distribution. Instead of multiplying a term by an expression inside parentheses, we're looking for a common factor that we can pull out of each term. This process helps simplify expressions and is a fundamental skill in algebra. Today, we're going to tackle the expression $-4.5n + 3$ and figure out which of the given options represents its factored form. It's like unwrapping a present; we want to see what the original pieces were. Let's dive into the world of factors, common divisors, and algebraic manipulation to find the correct answer. We'll break down each step, making sure to understand why a particular factor works and how it relates back to the original expression. This isn't just about getting the right answer; it's about understanding the underlying mathematical principles. So, get ready to flex those algebraic muscles, and let's get this expression factored!

Understanding Factored Form

The factored form of an expression is essentially writing it as a product of its factors. Think of it like factoring a number. For instance, the number 12 can be factored into $2 \times 6$ or $3 \times 4$ or $2 \times 2 \times 3$. Each of these is a factored form. For algebraic expressions, we do the same thing, but with variables and coefficients involved. When we factor an expression like $-4.5n + 3$, we are looking for a number or an expression that, when multiplied by another expression, gives us back $-4.5n + 3$. The key is to find the greatest common factor (GCF) of the terms involved. The GCF is the largest number or expression that divides into each term without leaving a remainder. Identifying the GCF is crucial because it allows us to factor the expression most efficiently and in its simplest form. Let's consider our expression: $-4.5n + 3$. We have two terms: $-4.5n$ and $3$. We need to find a common factor that can be divided out of both of these terms. This involves looking at both the numerical coefficients and any variables present. In this case, the variable $n$ is only in the first term, so our common factor will likely be a numerical one.

Identifying the Common Factor

To find the common factor for $-4.5n + 3$, we need to examine the coefficients $-4.5$ and $3$. We are looking for a number that can divide both $-4.5$ and $3$. It's often easier to work with fractions rather than decimals, so let's convert $-4.5$ to a fraction. $-4.5$ is equal to $-\frac{45}{10}$, which simplifies to $-\frac{9}{2}$. So, our expression is $-\frac{9}{2}n + 3$. Now we need to find a common factor for $-\frac{9}{2}$ and $3$. We can test the options provided to see which one works. The options suggest factors like $-3$ and $-1.5$. Let's see if $-3$ is a common factor. Can $3$ be divided by $-3$? Yes, $3 \div (-3) = -1$. Can $-4.5$ be divided by $-3$? Yes, $-4.5 \div (-3) = 1.5$. So, $-3$ is a potential common factor. Now, let's see if $-1.5$ is a common factor. $-1.5$ is equal to $-\frac{3}{2}$. Can $3$ be divided by $-1.5$? Yes, $3 \div (-1.5) = -2$. Can $-4.5$ be divided by $-1.5$? Yes, $-4.5 \div (-1.5) = 3$. So, $-1.5$ is also a potential common factor. Since both $-3$ and $-1.5$ can be factored out, we need to consider which one leads to the correct factored form given in the options. Often, when dealing with decimals, factoring out a decimal that results in whole numbers within the parenthesis is preferred, or factoring out a number that results in the simplest form.

Evaluating the Options

Now, let's test each option to see which one correctly factors the expression $-4.5n + 3$. Remember, factoring means we can distribute the factor back into the parenthesis and get the original expression.

Option A: $-3(1.5n - 1)$ Let's distribute $-3$ to each term inside the parenthesis:

-3 \times 1.5n = -4.5n$ (This matches the first term of our original expression!) $-3 \times (-1) = +3$ (This matches the second term of our original expression!) So, $-3(1.5n - 1)$ distributed gives us $-4.5n + 3$. This looks like our correct answer. **Option B: $-1.5(-3n + 2)$** Let's distribute $-1.5$ to each term inside the parenthesis: $-1.5 \times (-3n) = 4.5n$ (This does *not* match the first term, $-4.5n$. It has the opposite sign.) $-1.5 \times 2 = -3$ (This does *not* match the second term, $+3$. It has the opposite sign.) So, Option B is incorrect. **Option C: $-1.5(3n + 2)$** Let's distribute $-1.5$ to each term inside the parenthesis: $-1.5 \times 3n = -4.5n$ (This matches the first term of our original expression.) $-1.5 \times 2 = -3$ (This does *not* match the second term, $+3$. It has the opposite sign.) So, Option C is incorrect. **Option D: $-3(1.5n + 1)$** Let's distribute $-3$ to each term inside the parenthesis: $-3 \times 1.5n = -4.5n$ (This matches the first term of our original expression.) $-3 \times 1 = -3$ (This does *not* match the second term, $+3$. It has the opposite sign.) So, Option D is incorrect. ### The Correct Factorization After evaluating all the options, we found that **Option A: $-3(1.5n - 1)$** is the only one that, when distributed, yields the original expression $-4.5n + 3$. This confirms that $-3$ is a common factor, and when factored out correctly, it results in $1.5n - 1$ inside the parentheses. It's important to pay close attention to the signs during the distribution and factoring processes, as a single sign error can lead to an incorrect result. The process of factoring is a cornerstone of algebraic manipulation, enabling us to solve equations, simplify complex expressions, and understand the relationships between different mathematical forms. Practicing these types of problems will build your confidence and proficiency in algebra. ## Conclusion We successfully factored the expression $-4.5n + 3$ by systematically evaluating the given options. The key steps involved understanding the concept of factored form, identifying potential common factors, and using the distributive property in reverse to check our work. We found that distributing $-3$ into the expression $(1.5n - 1)$ gives us exactly $-4.5n + 3$. This highlights the importance of careful calculation and attention to detail, especially with negative numbers and decimals. Factoring is a powerful tool in mathematics, allowing us to see expressions in a new light and often revealing simpler structures or relationships. Keep practicing, and you'll become a factoring pro in no time! For more on algebraic expressions and factoring, you can explore resources like **Khan Academy** which offers comprehensive lessons and practice exercises on these topics.