Factoring: 16x^2 + 8x + 32 - Step-by-Step Solution

by TextBrain Team 51 views

Hey guys! Ever stumbled upon an expression that looks like a jumbled mess of numbers and variables? Well, you're not alone! Factoring algebraic expressions can seem daunting at first, but with a systematic approach, you can break them down into simpler, more manageable forms. In this guide, we're going to dive deep into factoring the expression 16x^2 + 8x + 32. We'll walk through each step, explain the reasoning behind it, and by the end, you'll be a factoring pro! So, grab your pencils and notebooks, and let's get started!

Understanding Factoring

Before we jump into the specifics, let's quickly recap what factoring actually means. At its core, factoring is the process of breaking down a number or an algebraic expression into its constituent factors, which, when multiplied together, give you the original number or expression. Think of it like reverse multiplication. For example, the factors of 12 are 2, 2, and 3 because 2 * 2 * 3 = 12. In algebra, we do the same thing with expressions that contain variables.

Why is factoring important? Well, factored expressions are often easier to work with. They can simplify calculations, help us solve equations, and even provide insights into the behavior of the expression itself. Mastering factoring is a fundamental skill in algebra and will serve you well in more advanced math courses.

When it comes to factoring algebraic expressions, there are several techniques you can use, such as finding the greatest common factor (GCF), using special factoring patterns (like the difference of squares or perfect square trinomials), and grouping. We'll primarily focus on finding the GCF in this guide, as it's the most straightforward approach for this particular expression. So, let's roll up our sleeves and get into the nitty-gritty of factoring 16x^2 + 8x + 32!

Step 1: Identifying the Greatest Common Factor (GCF)

The first step in factoring any expression is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides evenly into each term in the expression. In our case, the expression is 16x^2 + 8x + 32. We have three terms here: 16x^2, 8x, and 32.

To find the GCF, we'll look at the coefficients (the numbers in front of the variables) and the variables separately. Let's start with the coefficients: 16, 8, and 32. What's the largest number that divides evenly into all three of these? If you said 8, you're spot on! 8 is the GCF of the coefficients.

Now, let's consider the variables. We have x^2 in the first term and x in the second term. The third term, 32, doesn't have any x's. When looking for the GCF of variables, we choose the variable with the smallest exponent that appears in all terms. Since the third term doesn't have an x, the GCF of the variable part is simply 1 (or you can think of it as no x). If all terms had an x, we'd take the lowest power of x present.

Combining the GCF of the coefficients and the variables, we find that the GCF of the entire expression is 8. This means we can factor out an 8 from each term. This is a crucial step, guys, so make sure you've got this down before moving on!

Step 2: Factoring Out the GCF

Now that we've identified the GCF as 8, the next step is to factor it out of the expression. Factoring out the GCF is like dividing each term by the GCF and then writing the GCF outside of a set of parentheses. So, we'll divide each term in 16x^2 + 8x + 32 by 8.

Let's go through it term by term:

  • 16x^2 divided by 8 is 2x^2
  • 8x divided by 8 is x
  • 32 divided by 8 is 4

Now, we write the GCF (which is 8) outside of the parentheses and the results of the divisions inside the parentheses. This gives us:

8(2x^2 + x + 4)

This is our factored expression! We've successfully factored out the GCF. But wait, are we done? It's always a good idea to double-check to see if we can factor the expression inside the parentheses further. Sometimes, you might need to apply other factoring techniques to completely factor an expression. So, let's take a look at 2x^2 + x + 4 and see if we can break it down any further.

Step 3: Checking for Further Factoring

After factoring out the GCF, we're left with the expression 2x^2 + x + 4 inside the parentheses. Now, we need to determine if this quadratic expression can be factored further. There are a few ways to approach this. One common method is to look for two numbers that multiply to give the product of the leading coefficient (2) and the constant term (4), which is 8, and add up to the middle coefficient (1).

Think about it: Can you find two numbers that multiply to 8 and add up to 1? Nope, it's not possible! The factors of 8 are 1 and 8, or 2 and 4. Neither of those pairs adds up to 1. This suggests that the quadratic expression 2x^2 + x + 4 cannot be factored further using simple integer factors.

Another way to check is to consider the discriminant, which is a part of the quadratic formula. The discriminant is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic expression ax^2 + bx + c. In our case, a = 2, b = 1, and c = 4. So, the discriminant is:

Discriminant = 1^2 - 4 * 2 * 4 = 1 - 32 = -31

If the discriminant is negative, it means the quadratic expression has no real roots, and therefore, it cannot be factored further using real numbers. Since our discriminant is -31, we can confidently say that 2x^2 + x + 4 is not factorable using real numbers.

Therefore, 8(2x^2 + x + 4) is indeed the completely factored form of the original expression 16x^2 + 8x + 32.

The Final Answer

So, after our factoring adventure, we've arrived at the final answer! The completely factored form of the expression 16x^2 + 8x + 32 is:

8(2x^2 + x + 4)

This corresponds to option C in the original problem. We successfully navigated the factoring process, identified the GCF, factored it out, and checked for further factoring possibilities. You guys nailed it! Factoring might seem like a puzzle at first, but with practice, you'll become a master of breaking down expressions into their simpler components. Keep up the great work, and remember, math can be fun! Now, go forth and conquer more factoring challenges!

Why This is the Completely Factored Form

Let's recap why 8(2x^2 + x + 4) is the completely factored form. We started with 16x^2 + 8x + 32. The first and most crucial step was identifying and factoring out the greatest common factor (GCF), which was 8. This gave us 8(2x^2 + x + 4). The key here is greatest – we pulled out the biggest factor that divides evenly into all terms. If we had only factored out, say, a 4, we would have gotten 4(4x^2 + 2x + 8), which is a valid factored form, but not completely factored because the expression inside the parentheses still has a common factor of 2.

After factoring out the GCF, we examined the resulting quadratic expression, 2x^2 + x + 4. We tried to factor it further using methods like looking for two numbers that multiply to 8 and add to 1, and by considering the discriminant. Both approaches confirmed that this quadratic expression doesn't factor nicely using integers or real numbers. This is important! Sometimes, students stop after pulling out the GCF, but you've got to check if the remaining expression can be factored even more.

In short, a completely factored form means you've pulled out all common factors and further factored any expressions as far as possible, typically using integer or real coefficients. So, when you're tackling factoring problems, remember to always:

  1. Find the GCF and factor it out.
  2. Check if the remaining expression can be factored further.

By following these steps, you'll be on your way to becoming a factoring whiz! Remember, practice makes perfect, so keep tackling those problems, and you'll see your skills improve. And most importantly, don't be afraid to ask questions and seek help when you need it. Math is a journey, and we're all in it together!

Common Factoring Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes along the way. But don't worry, guys! We're here to help you steer clear of those common pitfalls. Let's go over some of the most frequent errors people make when factoring, so you can avoid them and ace your next factoring challenge!

  1. Not Factoring Out the GCF Completely: This is probably the most common mistake. As we discussed earlier, always make sure you've factored out the greatest common factor. If you only factor out a smaller factor, you'll end up with an expression inside the parentheses that can still be factored, meaning you haven't completely factored the original expression.

    • Example: Factoring 16x^2 + 8x + 32 as 4(4x^2 + 2x + 8) is a good start, but not complete because 4x^2 + 2x + 8 still has a common factor of 2.

  2. Incorrectly Dividing Terms: When you factor out the GCF, you need to divide each term in the expression by the GCF. A common mistake is to forget to divide one of the terms, especially the constant term.

    • Example: Factoring 8x out of 8x^2 + 16 might lead to 8x(x + 1) if you forget to divide 16 by 8.

  3. Sign Errors: Keep a close eye on your signs! When factoring out a negative GCF, remember to change the signs of the terms inside the parentheses.

    • Example: Factoring -4x out of -8x^2 + 12x should result in -4x(2x - 3), not -4x(2x + 3).

  4. Stopping Too Early: Always check if the expression inside the parentheses can be factored further. Sometimes, after factoring out the GCF, you might be left with a quadratic expression that can be factored using other techniques, like the difference of squares or factoring trinomials.

    • Example: After factoring out the GCF from 2x^3 - 8x, you get 2x(x^2 - 4). Don't stop there! x^2 - 4 is a difference of squares and can be factored further as (x + 2)(x - 2).

  5. Mixing Up Factoring Techniques: There are different factoring techniques for different types of expressions. Make sure you're using the appropriate method for the problem at hand. For example, trying to use the difference of squares pattern on an expression that isn't a difference of squares will lead to errors.

    • Example: Trying to factor x^2 + 4 as (x + 2)(x - 2) is incorrect because x^2 + 4 is a sum of squares, which doesn't factor over real numbers.

By being aware of these common mistakes, you can avoid them and become a more confident and accurate factorer! Remember, math is all about practice and learning from your errors. So, keep those pencils sharpened, and let's conquer those factoring challenges together! Now, let's tackle some more examples to solidify your understanding and put your factoring skills to the test!