Factoring: Find Factors Of 5(3x² + 9x) - 14
Hey guys! Today, we're diving deep into the world of factoring expressions. Factoring can seem daunting at first, but with a systematic approach, it becomes a breeze. We'll break down the expression 5(3x² + 9x) - 14, identify its factors, and equip you with the skills to tackle similar problems. Let's get started!
Understanding the Expression: 5(3x² + 9x) - 14
Before we jump into factoring, let's take a closer look at the expression 5(3x² + 9x) - 14. This expression is a polynomial, which simply means it's an expression containing variables and coefficients combined using addition, subtraction, and multiplication. In this case, we have a quadratic expression (because of the x² term) inside the parentheses, multiplied by a constant, and then a constant subtracted from the result.
To effectively factor this expression, we need to understand the order of operations and how to manipulate algebraic terms. The key is to simplify the expression first by distributing the 5 and then looking for common factors. This process will help us reveal the underlying structure of the expression and make it easier to identify its factors. Factoring, in essence, is the reverse of distribution; we're trying to find the building blocks that, when multiplied together, give us the original expression. Keep in mind that a factor is a number or expression that divides evenly into another number or expression. So, our goal is to find these building blocks for our given expression.
Step-by-Step Factoring Process
Now, let's break down the factoring process step by step:
1. Distribute the Constant
The first step is to distribute the 5 across the terms inside the parentheses. This means multiplying 5 by both 3x² and 9x:
5 * (3x²) = 15x² 5 * (9x) = 45x
So, after distributing, our expression becomes:
15x² + 45x - 14
This step is crucial because it removes the parentheses and allows us to see all the terms clearly. Distributing the constant simplifies the expression and prepares it for further factoring steps. It's like unwrapping a gift – we need to remove the outer layer to see what's inside!
2. Look for Common Factors
Next, we need to look for any common factors among the terms 15x², 45x, and -14. A common factor is a number or variable that divides evenly into all the terms. In this case, let's first look at the coefficients: 15, 45, and -14.
The greatest common divisor (GCD) of 15 and 45 is 15, but 15 does not divide evenly into -14. Therefore, there's no common numerical factor for all three terms. Now, let's consider the variable x. The first two terms, 15x² and 45x, have x as a common factor, but the last term, -14, does not have any x. This means there's no common variable factor for all three terms either.
Since we couldn't find any common factors for all three terms, this suggests that the expression 15x² + 45x - 14 might not be factorable using simple techniques like factoring out a common factor. This doesn't mean it's impossible to factor, but it does mean we might need to explore other methods, such as the quadratic formula or completing the square, if we were trying to find the roots of the equation. However, for the purpose of identifying factors in the expression itself, we've reached a point where we can conclude that there are no readily apparent common factors.
3. Analyzing the Options
Now that we've simplified the expression and looked for common factors, let's analyze the given options to see if any of them represent a factor of the expression 15x² + 45x - 14:
A. 45x B. 70 C. 5
Option A: 45x
45x is a term that appears in the expression after distributing (before subtracting 14), but it's not a factor of the entire expression 15x² + 45x - 14. A factor needs to divide the entire expression evenly, which 45x does not do because of the -14 term. Think of it this way: if you divide 15x² + 45x - 14 by 45x, you won't get a clean, whole expression as a result. There will be a remainder.
Option B: 70
70 is a constant, and while it might seem like a potential factor because it's a multiple of the constant term -14 (70 = -14 * -5), it's not a factor of the entire expression. The same logic applies here: dividing 15x² + 45x - 14 by 70 won't result in a clean expression. The variable terms (15x² and 45x) prevent 70 from being a factor.
Option C: 5
5 is a constant, and it was initially multiplied by the expression in parentheses. However, due to the subtraction of 14 outside the parentheses, 5 is not a factor of the entire expression 15x² + 45x - 14. The -14 term breaks the possibility of 5 being a factor of the whole expression. If the expression were simply 5(3x² + 9x), then 5 would definitely be a factor. But the presence of -14 changes the game.
Conclusion: Identifying Factors
After careful analysis, we can conclude that none of the given options (45x, 70, and 5) represent a factor of the expression 5(3x² + 9x) - 14 or its simplified form 15x² + 45x - 14. The key reason is that a factor must divide the entire expression evenly, and the presence of the constant term -14 prevents any of these options from doing so.
Remember, factoring is like reverse multiplication. We're looking for expressions that, when multiplied together, give us the original expression. In this case, the expression 15x² + 45x - 14 doesn't have any obvious factors among the options provided. This highlights an important point: not all expressions can be factored easily, and sometimes, they might not be factorable at all using simple techniques.
So, guys, the takeaway here is to always follow a systematic approach to factoring: distribute, look for common factors, and then carefully analyze the options. And remember, practice makes perfect! The more you work with factoring, the more comfortable and confident you'll become.
