Polynomial Division: Express Answers In Fraction Form

Nov 7, 2025 by Admin 54 views

Hey guys! Let's dive into the world of polynomial division, where we'll tackle a couple of problems and make sure to express our answers in fraction form. It might sound a bit intimidating, but trust me, we'll break it down step by step so it's super easy to follow. We’re going to solve these problems just like we would with regular numbers, but with a bit of algebraic flair. So, grab your pencils, and let's get started!

Dividing Polynomials: Problem 1

Let's kick things off with our first problem. We need to divide ${ y'(x) = 4x^3 - 2x^2 + x - 3 }$ by ${ d(x) = x - 1 }$ . So, in essence, we're trying to figure out what happens when we split the polynomial ${ 4x^3 - 2x^2 + x - 3 }$ into chunks the size of ${ x - 1 }$ . Sounds like a puzzle, right? But it's a fun one, I promise!

Setting Up the Division

First, we'll set up our division problem using the long division method, which is pretty similar to how you'd divide numbers. We'll write ${ 4x^3 - 2x^2 + x - 3 }$ inside the division bracket and ${ x - 1 }$ outside. Think of it like setting up a stage for our algebraic performance – everything needs to be in its place.

Performing the Division

Now comes the fun part – the actual division! We start by looking at the highest power of ${ x }$ in our dividend (the polynomial inside the bracket), which is ${ 4x^3 }$ . We ask ourselves, "What do we need to multiply ${ x }$ (from our divisor ${ x - 1 }$ ) by to get ${ 4x^3 }$ ?" The answer is ${ 4x^2 }$ . So, we write ${ 4x^2 }$ above the division bracket, aligned with the ${ x^2 }$ term.

Next, we multiply the entire divisor ${ x - 1 }$ by ${ 4x^2 }$ , which gives us ${ 4x^3 - 4x^2 }$ . We write this result below our dividend and subtract it. This step is crucial; it's like carefully carving away pieces to reveal the hidden solution.

After subtracting, we get ${ 2x^2 + x - 3 }$ . We bring down the next term ( ${ +x }$ ) from the dividend. Now, we repeat the process. What do we need to multiply ${ x }$ by to get ${ 2x^2 }$ ? The answer is ${ 2x }$ . We write ${ +2x }$ above the division bracket, next to ${ 4x^2 }$ .

Multiply ${ 2x }$ by ${ x - 1 }$ to get ${ 2x^2 - 2x }$ . Subtract this from ${ 2x^2 + x }$ to get ${ 3x - 3 }$ . Bring down the last term, ${ -3 }$ , and we have ${ 3x - 3 }$ . This feels like we're peeling away layers, doesn't it?

One last time! What do we multiply ${ x }$ by to get ${ 3x }$ ? The answer is ${ 3 }$ . Write ${ +3 }$ above the division bracket. Multiply ${ 3 }$ by ${ x - 1 }$ to get ${ 3x - 3 }$ . Subtract this, and we get ${ 0 }$ . Woohoo! No remainder.

The Result

So, the result of dividing ${ 4x^3 - 2x^2 + x - 3 }$ by ${ x - 1 }$ is ${ 4x^2 + 2x + 3 }$ . Since there's no remainder, we can say that ${ x - 1 }$ divides evenly into ${ 4x^3 - 2x^2 + x - 3 }$ . That's one problem down!

Diving Polynomials: Problem 2

Now, let’s tackle our second problem. This time, we need to divide ${ f(x) = 3x^4 - 3x^3 - 9x^2 + 3x - 2 }$ by ${ d(x) = x - 2 }$ . This one looks a bit longer, but don’t worry, we’ve got this! We'll use the same long division method as before, breaking it down step by step.

Setting Up the Division

Just like before, we set up our long division. We write ${ 3x^4 - 3x^3 - 9x^2 + 3x - 2 }$ inside the division bracket and ${ x - 2 }$ outside. It’s all about setting the stage correctly, guys.

Performing the Division

Alright, let’s get to the division. We start with the highest power of ${ x }$ in our dividend, which is ${ 3x^4 }$ . We ask, "What do we need to multiply ${ x }$ by to get ${ 3x^4 }$ ?" The answer is ${ 3x^3 }$ . So, we write ${ 3x^3 }$ above the division bracket, aligning it with the ${ x^3 }$ term.

Multiply ${ 3x^3 }$ by ${ x - 2 }$ to get ${ 3x^4 - 6x^3 }$ . We write this below our dividend and subtract it. This subtraction step is super important for keeping everything neat and tidy.

After subtracting, we get ${ 3x^3 - 9x^2 + 3x - 2 }$ . We bring down the next term ( ${ -9x^2 }$ ) from the dividend. Now, we repeat the process. What do we need to multiply ${ x }$ by to get ${ 3x^3 }$ ? The answer is ${ 3x^2 }$ . We write ${ +3x^2 }$ above the division bracket, next to ${ 3x^3 }$ .

Multiply ${ 3x^2 }$ by ${ x - 2 }$ to get ${ 3x^3 - 6x^2 }$ . Subtract this from ${ 3x^3 - 9x^2 }$ to get ${ -3x^2 + 3x - 2 }$ . Bring down the next term, ${ +3x }$ , and we have ${ -3x^2 + 3x - 2 }$ .

Let’s keep going! What do we multiply ${ x }$ by to get ${ -3x^2 }$ ? The answer is ${ -3x }$ . Write ${ -3x }$ above the division bracket. Multiply ${ -3x }$ by ${ x - 2 }$ to get ${ -3x^2 + 6x }$ . Subtract this, and we get ${ -3x - 2 }$ .

One last step! Bring down the ${ -2 }$ , so we have ${ -3x - 2 }$ . What do we multiply ${ x }$ by to get ${ -3x }$ ? The answer is ${ -3 }$ . Write ${ -3 }$ above the division bracket. Multiply ${ -3 }$ by ${ x - 2 }$ to get ${ -3x + 6 }$ . Subtract this from ${ -3x - 2 }$ to get ${ -8 }$ . Ah, this time we have a remainder!

The Result

So, the result of dividing ${ 3x^4 - 3x^3 - 9x^2 + 3x - 2 }$ by ${ x - 2 }$ is ${ 3x^3 + 3x^2 - 3x - 3 }$ with a remainder of ${ -8 }$ . To express this in fraction form, we write the remainder over the divisor:

${ 3x^3 + 3x^2 - 3x - 3 - \frac{8}{x - 2} }$

And there we have it! We’ve successfully divided the polynomial and expressed our answer in fraction form. See, it’s totally doable!

Key Steps in Polynomial Long Division

Let’s recap the key steps we used in polynomial long division. This will help solidify the process in your mind and make sure you’re ready to tackle any division problem that comes your way.

Set Up the Division: Write the dividend (the polynomial being divided) inside the division bracket and the divisor (the polynomial you’re dividing by) outside. Make sure both polynomials are written in descending order of powers.
Divide the Leading Terms: Divide the highest power term of the dividend by the highest power term of the divisor. Write the result above the division bracket, aligned with the term of the same power.
Multiply: Multiply the result from step 2 by the entire divisor. Write the product below the dividend, aligning terms of the same power.
Subtract: Subtract the product from the dividend. Be careful to distribute the negative sign correctly.
Bring Down: Bring down the next term from the dividend and add it to the result of the subtraction.
Repeat: Repeat steps 2-5 until all terms from the dividend have been brought down and divided.
Remainder: If there is a remainder, write it as a fraction over the divisor.

Tips and Tricks for Polynomial Division

Polynomial division can be a bit tricky at first, but with a few tips and tricks, you'll become a pro in no time. Here are some things to keep in mind:

Keep it Organized: Make sure to align terms of the same power when you're writing out the division. This will help you avoid mistakes and keep your work neat and tidy.
Watch the Signs: Pay close attention to the signs when you're subtracting. A small mistake with a sign can throw off your entire answer.
Fill in Missing Terms: If a polynomial is missing a term (for example, if it goes from ${ x^3 }$ to ${ x }$ without an ${ x^2 }$ term), you can add a placeholder with a coefficient of 0 (like ${ 0x^2 }$ ). This helps keep your columns aligned.
Check Your Work: After you've finished the division, you can check your answer by multiplying the quotient by the divisor and adding the remainder. If you get the original dividend, you know you've done it right.

Expressing the Remainder in Fraction Form

One of the key things we focused on in these problems is expressing the remainder in fraction form. When you have a remainder after polynomial division, you write it as a fraction with the remainder as the numerator and the divisor as the denominator.

For example, if you divide ${ P(x) }$ by ${ D(x) }$ and get a quotient ${ Q(x) }$ and a remainder ${ R(x) }$ , you can write the result as:

${ \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} }$

This is super important because it gives us a complete and accurate representation of the division result. It's like saying, "We divided as much as we could, and this is what’s left over." Plus, it’s the standard way to express these kinds of results in algebra.

Practice Makes Perfect

Like with any math skill, practice is the key to mastering polynomial division. The more problems you work through, the more comfortable you'll become with the process. Start with simpler problems and gradually work your way up to more complex ones.

Try making up your own problems, or find some in your textbook or online. Work through them step by step, and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise!

Conclusion

So, there you have it! We’ve tackled polynomial division and learned how to express our answers in fraction form. Remember, it’s all about breaking the problem down into manageable steps and staying organized. With a bit of practice, you’ll be dividing polynomials like a pro!

Keep practicing, and don't hesitate to ask for help if you get stuck. You’ve got this, guys! Polynomial division might seem tough at first, but you’ve taken a big step towards mastering it today. Happy dividing!