Hey everyone! Today, we're diving into a super interesting topic in algebra: polynomial division. Specifically, we're going to tackle how to divide the polynomial 5x^2 + 20x + 32 by the binomial x + 2. This might sound a bit intimidating at first, but trust me, guys, once you get the hang of the steps, it's actually pretty straightforward. We'll break it down step-by-step, making sure you understand each part of the process. So, grab your notebooks, maybe a cup of coffee, and let's get this done!

Understanding the Problem: What Are We Actually Doing?

Alright, so what does it mean to divide 5x^2 + 20x + 32 by x + 2? Think of it like regular division, where you have a dividend (the number being divided) and a divisor (the number you're dividing by). In our case, the dividend is 5x^2 + 20x + 32, and the divisor is x + 2. Our goal is to find out what we get when we perform this division. We're looking for a quotient (the result of the division) and potentially a remainder (what's left over). This process is a fundamental skill in algebra, essential for simplifying expressions, solving equations, and even graphing functions. It's like being a mathematical detective, figuring out how one polynomial fits into another. The terms we're dealing with are called polynomials, which are just expressions with one or more terms involving variables raised to non-negative integer powers. Here, 5x^2 + 20x + 32 is a quadratic polynomial (degree 2), and x + 2 is a linear polynomial (degree 1). When we divide a polynomial of degree 'm' by a polynomial of degree 'n', the resulting quotient will have a degree of 'm-n', and the remainder will have a degree less than 'n'. In our case, we expect a quotient with degree 2-1 = 1, and a remainder with degree less than 1 (meaning it's a constant or zero). This is a key concept to keep in mind as we work through the division. Don't worry if these terms sound a bit technical; we'll make them super clear as we go along. The structure of polynomials and how they relate to each other through division is a cornerstone of higher-level math, so understanding this now will set you up for success down the line. It's all about breaking down complex problems into manageable steps, and polynomial division is a perfect example of that.

Method 1: Polynomial Long Division - The Classic Approach

Okay, guys, let's roll up our sleeves and get into the nitty-gritty of polynomial long division. This method is probably the most familiar if you've done any kind of long division before, just with algebraic terms instead of plain numbers. First things first, we need to set up the problem. Write it out like you would a standard long division problem: the dividend (5x^2 + 20x + 32) goes inside the 'house', and the divisor (x + 2) goes outside. It's crucial that both the dividend and the divisor are written in descending order of powers. If any terms are missing (like if we didn't have an 'x' term), we'd add a placeholder with a coefficient of zero. In our case, 5x^2 + 20x + 32 is already in order, and x + 2 is too. So, we're good to go.

Step 1: Divide the leading terms. Look at the first term of the dividend (5x^2) and the first term of the divisor (x). Ask yourself: 'What do I need to multiply x by to get 5x^2?' The answer is 5x. Write this 5x above the 20x term in the quotient area. This is the first part of our answer.

Step 2: Multiply the result by the divisor. Now, take the 5x you just found and multiply it by the entire divisor (x + 2). So, 5x * (x + 2) = 5x^2 + 10x. Write this result directly below the dividend, aligning terms with the same powers.

Step 3: Subtract. This is where people sometimes make mistakes, so pay attention! Subtract the expression you just got (5x^2 + 10x) from the dividend. Remember to distribute the negative sign: (5x^2 + 20x) - (5x^2 + 10x) = 5x^2 + 20x - 5x^2 - 10x. This simplifies to 10x. Bring down the next term from the dividend (+ 32) so you now have 10x + 32.

Step 4: Repeat the process. Now, we treat 10x + 32 as our new dividend. Repeat Step 1: Divide the leading term of the new dividend (10x) by the leading term of the divisor (x). What do you multiply x by to get 10x? That's + 10. Write this + 10 in the quotient area, next to the 5x.

Step 5: Multiply and subtract again. Multiply this + 10 by the divisor (x + 2): 10 * (x + 2) = 10x + 20. Subtract this from 10x + 32: (10x + 32) - (10x + 20) = 10x + 32 - 10x - 20 = 12.

Step 6: Identify the remainder. Since 12 has a lower degree than the divisor (x + 2), we can't divide any further. So, 12 is our remainder.

Our final answer, expressed using the quotient and remainder, is 5x + 10 with a remainder of 12. We can also write this as 5x + 10 + 12/(x + 2). See? Not so scary after all!

Method 2: Synthetic Division - The Speedy Shortcut

Alright, fellow math enthusiasts, now let's talk about a method that's often quicker and less prone to sign errors: synthetic division. This is a fantastic shortcut, but it only works when you're dividing by a linear binomial of the form (x - c). In our case, we're dividing by (x + 2), which we can rewrite as (x - (-2)). So, the value of 'c' we'll be using is -2.

Step 1: Set up the synthetic division. Forget the long division house for a minute. Draw a small division bracket or a 'L' shape. On the top left, put the value of 'c' (which is -2 in our case). Then, list the coefficients of the dividend (5x^2 + 20x + 32) in a row to the right of the 'c'. Make sure your polynomial is in descending order and includes placeholders (with zero coefficients) for any missing terms. Our coefficients are 5, 20, and 32.

So, your setup will look something like this:

-2 | 5   20   32
   |___________

Step 2: Bring down the first coefficient. Take the first coefficient of the dividend (which is 5) and bring it straight down below the line. This number is the start of our quotient.

-2 | 5   20   32
   |___________
     5

Step 3: Multiply and add. Now, take the number you just brought down (5) and multiply it by 'c' (-2). 5 * (-2) = -10. Write this result under the next coefficient (20).

-2 | 5   20   32
   |  -10
   |___________
     5

After multiplying, add the numbers in that column (20 + (-10) = 10). Write this sum below the line.

-2 | 5   20   32
   |  -10
   |___________
     5   10

Step 4: Repeat the multiply and add process. Take the new number below the line (10) and multiply it by 'c' (-2). 10 * (-2) = -20. Write this result under the next coefficient (32).

-2 | 5   20   32
   |  -10  -20
   |___________
     5   10

Add the numbers in this column (32 + (-20) = 12). Write this sum below the line.

-2 | 5   20   32
   |  -10  -20
   |___________
     5   10   12

Step 5: Interpret the results. The numbers below the line, except for the last one, are the coefficients of your quotient. The last number is the remainder. Since our original dividend was a quadratic (degree 2) and we divided by a linear term (degree 1), our quotient will be a linear term (degree 1). So, the 5 and 10 are the coefficients of our quotient.

This means our quotient is 5x + 10. The last number, 12, is the remainder.

So, just like with long division, the result is 5x + 10 with a remainder of 12, or 5x + 10 + 12/(x + 2). See how much faster that was? Synthetic division is a real lifesaver for these types of problems!

Checking Your Work: Does It All Add Up?