Polynomial Division: Solving (x^2-3x+9) / (x-2)

Nov 10, 2025 by Admin 48 views

Hey guys! Today, we're diving into polynomial division, and we're going to tackle a specific problem that might seem a bit tricky at first. But trust me, we'll break it down step by step so you can master it. Our mission is to divide the polynomial ${x^2 - 3x + 9}$ by ${x - 2}$ and express the result in a special form: ${p(x) + \frac{k}{x-2}}$ . Here, ${p(x)}$ represents another polynomial, and ${k}$ is just a plain old integer. Sounds like fun, right? Let’s get started!

Understanding Polynomial Division

Before we jump into the actual problem, let's quickly recap what polynomial division is all about. Think of it like regular long division, but instead of numbers, we're dealing with expressions involving variables (like ${x}$ ). The goal is the same: to figure out how many times one polynomial (the divisor) fits into another (the dividend). When performing polynomial division, you're essentially trying to break down a more complex polynomial into simpler parts. This is super useful in various areas of mathematics, including finding roots of polynomials, simplifying expressions, and even in calculus. The key idea behind polynomial division is to systematically reduce the degree of the dividend until you're left with a remainder that has a lower degree than the divisor. Just like with numerical long division, we focus on matching terms and subtracting multiples until we can't divide any further. This process helps us rewrite rational functions in a form that's often easier to work with, particularly when we need to integrate or analyze the function's behavior. So, whether you're facing a homework problem or tackling a more advanced mathematical concept, mastering polynomial division is a valuable skill to have in your toolkit.

Why This Form Matters

You might be wondering, why do we need to express our answer in the form ${p(x) + \frac{k}{x-2}}$ ? Well, this form is particularly useful because it separates the polynomial part ( ${p(x)}$ ) from the remainder part ( ${\frac{k}{x-2}}$ ). This can make it easier to analyze the behavior of the original expression, especially when dealing with limits or graphing. Plus, it's a common way to express the result of polynomial division, so getting comfortable with it is a smart move.

Step-by-Step Solution

Okay, let’s get our hands dirty and solve the problem. We're going to use the long division method, which is a classic technique for dividing polynomials.

1. Set Up the Long Division

First, we set up the long division just like you would with numbers. The polynomial we're dividing ( ${x^2 - 3x + 9}$ ) goes inside the division symbol, and the polynomial we're dividing by ( ${x - 2}$ ) goes outside. It should look something like this:

 x - 2 | x^2 - 3x + 9

This setup mirrors the way we approach numerical long division, making the process more familiar and less daunting. When setting up, it's important to make sure both the divisor and the dividend are written in descending order of powers of ${x}$ . This helps keep the process organized and reduces the chances of making mistakes. Also, if any powers of ${x}$ are missing in the dividend (e.g., if there's no ${x}$ term), it's a good idea to include them with a coefficient of zero. This can help maintain proper alignment during the division process. Remember, a clear setup is half the battle when it comes to long division, so take your time and make sure everything is in its place before moving on.

2. Divide the Leading Terms

Now, we focus on the leading terms. We ask ourselves: what do we need to multiply ${x}$ (from the divisor) by to get ${x^2}$ (the leading term of the dividend)? The answer is ${x}$ . So, we write ${x}$ above the division symbol, aligning it with the ${x}$ term in the dividend.

 x
 x - 2 | x^2 - 3x + 9

The leading term is the term with the highest power of the variable, and it's our focus in each step of the long division process. By targeting the leading terms, we ensure that we're systematically reducing the degree of the dividend. This is crucial for eventually arriving at a remainder that has a lower degree than the divisor, which is our goal. The alignment of terms is also key; placing the ${x}$ term above the ${-3x}$ term helps keep the calculation organized and reduces the risk of errors. As we move through the division, we'll continue to focus on the leading terms of the remaining polynomial, making the process manageable and efficient. Remember, each step builds on the previous one, so precision here sets the stage for a smooth and accurate solution.

3. Multiply and Subtract

Next, we multiply the ${x}$ we just wrote by the entire divisor ( ${x - 2}$ ): ${x * (x - 2) = x^2 - 2x}$ . We write this result below the dividend and subtract it. Make sure to align like terms!

 x
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)

 x
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)
 ---------
 -x

This step is where the core of the division process happens. Multiplying the term we placed above the division symbol by the entire divisor allows us to create a polynomial that, when subtracted from the dividend, will eliminate the leading term. This is how we reduce the degree of the polynomial we're working with. The subtraction itself is crucial, and it's important to distribute the negative sign correctly to each term. This is a common spot for errors, so double-check your work here. Aligning like terms is also vital; it ensures that you're subtracting the correct coefficients from each other. Once we've subtracted, we bring down the next term from the dividend to continue the process. This cycle of dividing, multiplying, subtracting, and bringing down terms is the heart of polynomial long division, and mastering it will make these problems much more manageable.

4. Bring Down the Next Term

Bring down the ${+9}$ from the dividend.

 x
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)
 ---------
 -x + 9

Bringing down the next term is a simple but important step in the long division process. It keeps the problem moving forward and ensures that we're considering all parts of the original dividend. Think of it like bringing down the next digit in numerical long division. This step sets us up for the next round of dividing, multiplying, and subtracting. By bringing down the ${+9}$ , we now have a new polynomial, ${-x + 9}$ , to work with. This process helps us systematically reduce the complexity of the dividend until we reach a remainder, making the division manageable one step at a time. So, don't forget this step – it's essential for keeping the long division on track!

5. Repeat the Process

Now we repeat the process. What do we need to multiply ${x}$ (from the divisor) by to get ${-x}$ ? The answer is ${-1}$ . Write ${-1}$ above the division symbol.

 x - 1
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)
 ---------
 -x + 9

Multiply ${-1}$ by the divisor ${x - 2}$ : ${-1 * (x - 2) = -x + 2}$ . Write this below ${-x + 9}$ and subtract.

 x - 1
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)
 ---------
 -x + 9
 -(-x + 2)

 x - 1
 x - 2 | x^2 - 3x + 9
 -(x^2 - 2x)
 ---------
 -x + 9
 -(-x + 2)
 ---------
 7

Repeating the process is where the rhythm of long division really comes into play. We're essentially doing the same steps over and over until we can't divide any further. The key is to stay focused on the leading terms and make sure each multiplication and subtraction is accurate. Finding the right term to multiply the divisor by is crucial for reducing the polynomial. Then, multiplying and subtracting correctly eliminates the leading term, allowing us to bring down the next term and continue the cycle. This iterative process is what makes long division such a powerful tool for breaking down complex polynomials into simpler parts. So, take each step deliberately, and you'll find that this repetition becomes almost second nature.

6. Write the Remainder

We're left with a remainder of ${7}$ . Since the degree of the remainder (which is 0, because 7 is a constant) is less than the degree of the divisor (which is 1, because ${x - 2}$ is a linear term), we can't divide any further.

7. Express the Answer

Now we can express our answer in the form ${p(x) + \frac{k}{x-2}}$ . Our quotient (the polynomial above the division symbol) is ${x - 1}$ , and our remainder is ${7}$ . So, the answer is:

${ x - 1 + \frac{7}{x - 2} }$

And that’s it! We've successfully divided the polynomials and expressed the result in the required form.

Putting It All Together

So, to recap, we divided ${x^2 - 3x + 9}$ by ${x - 2}$ using long division. We followed these steps:

Set up the long division.
Divided the leading terms.
Multiplied and subtracted.
Brought down the next term.
Repeated the process until we couldn't divide any further.
Expressed the answer in the form ${p(x) + \frac{k}{x-2}}$ .

Our final answer was ${x - 1 + \frac{7}{x - 2}}$ .

Practice Makes Perfect

Polynomial division might seem a bit daunting at first, but like anything else, practice makes perfect. Try working through a few more examples on your own, and you'll get the hang of it in no time. Remember, the key is to break it down step by step and stay organized. You got this!

Conclusion

Dividing polynomials is a fundamental skill in algebra, and mastering it opens doors to more advanced topics. By understanding the process and practicing regularly, you'll become more confident in your mathematical abilities. Plus, it's kind of satisfying to solve these problems, right? Keep up the great work, and I'll see you in the next math adventure!