Polynomial Division Step-by-Step Guide To Finding H(x) And S(x)

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Hey guys! Ever wrestled with polynomial division? It can seem tricky, but once you get the hang of it, it's actually pretty cool. Let's break down some problems step-by-step and make sure you understand how to find the quotient, h(x), and the remainder, s(x), when dividing polynomials. We'll tackle three examples that cover different scenarios, so you'll be a polynomial division pro in no time!

1. Dividing f(x) = 4x³ + 7x² - 18x + 8 by (4x - 1)

Let's dive straight into our first problem. We're given the polynomial f(x) = 4x³ + 7x² - 18x + 8 and we need to divide it by (4x - 1). Our goal is to find the quotient, which we'll call h(x), and the remainder, s(x). Polynomial division is much like long division with numbers, but instead of digits, we're working with terms containing x. So, let's get started and find out how to break this down.

First off, we set up our long division problem. Write 4x³ + 7x² - 18x + 8 inside the division symbol and (4x - 1) outside. Now, we focus on the leading terms. What do we need to multiply (4x) by to get 4x³? The answer is x². So, we write x² above the division symbol, in the x² column. Next, multiply the entire divisor, (4x - 1), by x². This gives us 4x³ - x². We write this result below our original polynomial and subtract. (4x³ + 7x²) - (4x³ - x²) equals 8x². Bring down the next term from our original polynomial, which is -18x. Now we have 8x² - 18x. What do we need to multiply (4x) by to get 8x²? The answer is 2x. Write +2x next to x² above the division symbol. Multiply (4x - 1) by 2x, which gives us 8x² - 2x. Subtract this from 8x² - 18x, and we get -16x. Bring down the last term, +8. Now we have -16x + 8. What do we multiply (4x) by to get -16x? The answer is -4. Write -4 next to 2x above the division symbol. Multiply (4x - 1) by -4, which gives us -16x + 4. Subtract this from -16x + 8, and we get a remainder of 4. So, after all that, we've found that h(x) = x² + 2x - 4 and s(x) = 4. Long division might seem like a lot of steps, but each step is straightforward, and with a little practice, you'll get super fast at it. Just remember to take it term by term and keep everything aligned properly.

So, to summarize, when f(x) = 4x³ + 7x² - 18x + 8 is divided by (4x - 1), the quotient h(x) is x² + 2x - 4, and the remainder s(x) is 4. Polynomial division is one of those foundational skills in algebra that you'll use again and again, so mastering it now will definitely pay off. Remember, the key is to break it down step by step, focusing on the leading terms and making sure your signs are correct. With practice, you'll be dividing polynomials like a pro!

2. Dividing f(x) = 4x³ + 7x² - 18x + 8 by (x² + 3x - 4)

Alright, let's tackle the second problem. This time we're dividing the same polynomial, f(x) = 4x³ + 7x² - 18x + 8, but our divisor is now a quadratic, (x² + 3x - 4). Don't worry, the principle is the same, but we need to be a little more careful with the terms we're dealing with. It's like going from simple division to multi-digit division – a few more steps, but totally manageable.

Just like before, we set up the long division. Put 4x³ + 7x² - 18x + 8 inside the division symbol and (x² + 3x - 4) outside. Now, let's think about the leading terms again. What do we need to multiply x² by to get 4x³? The answer is 4x. So, we write 4x above the division symbol, in the x column. Now, we multiply the entire divisor, (x² + 3x - 4), by 4x. This gives us 4x³ + 12x² - 16x. Write this below our original polynomial and subtract. (4x³ + 7x² - 18x) - (4x³ + 12x² - 16x) equals -5x² - 2x. Bring down the next term, +8, and we have -5x² - 2x + 8. Next up, what do we need to multiply x² by to get -5x²? The answer is -5. Write -5 next to 4x above the division symbol. Multiply (x² + 3x - 4) by -5, which gives us -5x² - 15x + 20. Subtract this from -5x² - 2x + 8. That's (-5x² - 2x + 8) - (-5x² - 15x + 20), which simplifies to 13x - 12. Now, can we divide (x² + 3x - 4) into (13x - 12)? Nope, because the degree of (13x - 12) (which is 1) is less than the degree of (x² + 3x - 4) (which is 2). This means (13x - 12) is our remainder. We've done it! So, the quotient h(x) is 4x - 5, and the remainder s(x) is 13x - 12.

Dividing by a quadratic adds a little twist, but the method is still the same. Keep focusing on those leading terms, make sure your subtraction is spot on, and you'll be golden. Remember, practice is the real game-changer here. The more you do, the quicker and more confident you'll become. And, hey, it's pretty satisfying when you nail a tricky division problem! So, don't shy away from these kinds of problems; embrace the challenge, and you'll level up your algebra skills in no time.

3. Dividing f(x) = 6x³ - x² - 15x + 8 by (2x² - 8x + 1)

Okay, let's jump into our final polynomial division problem. This one looks a bit more complex, but don't sweat it, we've got this! We're dividing f(x) = 6x³ - x² - 15x + 8 by (2x² - 8x + 1). The key here is to stay organized and follow the same steps we've been using. Let's see how it goes.

As always, we begin by setting up the long division. Write 6x³ - x² - 15x + 8 inside the division symbol and (2x² - 8x + 1) outside. Now, let’s focus on those leading terms. What do we need to multiply 2x² by to get 6x³? The answer is 3x. So, we write 3x above the division symbol, in the x column. Now, we multiply the entire divisor, (2x² - 8x + 1), by 3x. This gives us 6x³ - 24x² + 3x. Write this result below our original polynomial and subtract. (6x³ - x² - 15x) - (6x³ - 24x² + 3x) equals 23x² - 18x. Bring down the next term, +8, and we have 23x² - 18x + 8. Next, what do we need to multiply 2x² by to get 23x²? The answer is 23/2 (or 11.5 if you prefer decimals, but fractions are often easier to work with here). Write +23/2 next to 3x above the division symbol. Multiply (2x² - 8x + 1) by 23/2. This gives us 23x² - 92x + 23/2. Now, subtract this from 23x² - 18x + 8. It’s a bit messier this time: (23x² - 18x + 8) - (23x² - 92x + 23/2) simplifies to 74x - 7/2. Now, can we divide (2x² - 8x + 1) into (74x - 7/2)? Nope, because the degree of (74x - 7/2) (which is 1) is less than the degree of (2x² - 8x + 1) (which is 2). So, (74x - 7/2) is our remainder. We’ve conquered another one! So, for this problem, the quotient h(x) is 3x + 23/2, and the remainder s(x) is 74x - 7/2.

See, even with the fractions, the core process is the same. The most important thing is to keep everything aligned and take your time with the calculations. Fractions might seem intimidating, but they're just numbers too, so don't let them throw you off. Remember, math is like building with blocks – each step builds on the previous one. If you've got a solid foundation in the basics, you can tackle even the trickiest problems. And if you get stuck, don't be afraid to break the problem down into smaller parts or seek help from a teacher or friend. Keep practicing, and you'll be a polynomial division master in no time!

Key Takeaways for Polynomial Division

Alright, guys, let's recap what we've learned about polynomial division. We've tackled three different problems, each with its own little twist, but the core principles remain the same. Here's a quick rundown of the key things to remember when you're dividing polynomials:

  1. Set up the Long Division: Write the polynomial you're dividing (the dividend) inside the division symbol and the polynomial you're dividing by (the divisor) outside.
  2. Focus on Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. This gives you the first term of your quotient, h(x). Write this term above the division symbol, in the correct column.
  3. Multiply and Subtract: Multiply the entire divisor by the term you just wrote in the quotient. Write the result below the dividend and subtract. Remember to change the signs carefully when you subtract!
  4. Bring Down the Next Term: Bring down the next term from the dividend and write it next to the result of your subtraction.
  5. Repeat: Repeat steps 2-4 until you've brought down all the terms from the dividend.
  6. The Remainder: If the degree of the remaining polynomial is less than the degree of the divisor, then this is your remainder, s(x).
  7. Check Your Work: You can check your answer by multiplying the quotient, h(x), by the divisor and adding the remainder, s(x). The result should be the original dividend, f(x). This is just like checking regular long division with numbers!

Polynomial division is a fundamental skill in algebra, and it pops up in all sorts of places, from simplifying expressions to solving equations. So, the more comfortable you are with it, the better. And, like any math skill, the key to mastering polynomial division is practice, practice, practice! Don't be afraid to work through lots of examples, and don't get discouraged if you make mistakes along the way. Mistakes are just opportunities to learn and grow. Keep at it, and you'll become a polynomial division pro before you know it. Happy dividing!

Practice Problems

To help you solidify your understanding of polynomial division, here are a few practice problems you can try. Work through them step-by-step, using the methods we've discussed, and check your answers. Practice is the key to mastering any mathematical skill, so grab a pencil and paper and let's get to it!

  1. Divide f(x) = 2x³ - 5x² + 8x - 7 by (x - 2).
  2. Divide f(x) = x⁴ + 3x³ - 2x² + 5x - 1 by (x² + x - 1).
  3. Divide f(x) = 3x³ + 7x² - 4x + 2 by (3x - 2).

Remember to set up the long division carefully, focus on the leading terms, and take your time with the calculations. Check your answers by multiplying the quotient by the divisor and adding the remainder – the result should be the original dividend. Good luck, and happy dividing!

Conclusion

Alright, guys, we've covered a lot about polynomial division in this article. We started with the basics, worked through some examples, and even tackled a few tricky problems with fractions. We've also discussed some key takeaways and provided practice problems to help you hone your skills. So, where do we go from here? The most important thing is to keep practicing! Polynomial division might seem daunting at first, but with consistent effort, you'll become more confident and proficient. Don't be afraid to seek out additional resources, such as textbooks, online tutorials, or help from a teacher or tutor, if you're struggling with any of the concepts. Remember, mastering polynomial division is an investment in your mathematical future. It's a skill that will serve you well in algebra, calculus, and beyond. So, keep practicing, keep learning, and most importantly, keep having fun with math! And with that, I'm wrapping this article up. Thanks for joining me on this polynomial division journey. Until next time, happy problem-solving!