Long Division: Find Quotient, Remainder, And Factor

Hey guys! Today, we're diving into a classic math problem: long division with polynomials. It might seem a bit intimidating at first, but trust me, we'll break it down step by step. We're going to tackle a specific problem where we need to divide a polynomial by another polynomial, find the quotient and remainder, and then determine if the divisor is a factor of the dividend. So, let's get started and make math a little less mysterious!

Understanding Polynomial Long Division

Before we jump into the problem, let's quickly recap what polynomial long division actually is. Think of it like regular long division with numbers, but instead of digits, we're dealing with terms containing variables and exponents. The goal is the same: to divide one polynomial (the dividend) by another (the divisor) and find the quotient (the result of the division) and the remainder (what's left over). Understanding this process is key to solving the problem effectively.

The Process of Polynomial Long Division

Polynomial long division might seem complex, but it's actually a systematic process. You're essentially trying to figure out how many times one polynomial fits into another. Here’s a breakdown of the steps involved, which we’ll then apply to our specific problem:

1. Set up the division: Write the dividend inside the long division symbol and the divisor outside. Make sure both polynomials are written in descending order of exponents.
2. Divide the first terms: Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.
3. Multiply: Multiply the entire divisor by the first term of the quotient.
4. Subtract: Subtract the result from the dividend. This gives you a new polynomial.
5. Bring down the next term: Bring down the next term from the original dividend and add it to the new polynomial.
6. Repeat: Repeat steps 2-5 until there are no more terms to bring down.
7. Determine the remainder: The final polynomial left after the last subtraction is the remainder. If the remainder is zero, the divisor is a factor of the dividend. If the remainder is not zero, then the divisor is not a factor of the dividend. This is the critical part for our problem.

Why Long Division Matters

Polynomial long division isn't just a math exercise; it's a powerful tool. It helps us simplify complex polynomial expressions, factor polynomials, and solve equations. In many higher-level math courses, like calculus and abstract algebra, understanding polynomial division is crucial. Plus, it’s a great way to sharpen your algebraic skills. So, mastering this now will definitely pay off later!

Solving the Long Division Problem: A Step-by-Step Guide

Now, let's tackle the problem at hand. We need to divide $25y^3 - 25y^2 - 9y - 29$ by $5y - 2$. Grab your pencils, guys, and let's walk through it together.

Setting Up the Problem

First, let's set up the long division. Write $25y^3 - 25y^2 - 9y - 29$ inside the division symbol and $5y - 2$ outside. Make sure the terms are in descending order of their exponents, which they already are in this case. This is the foundation for a smooth division process. Now we are ready to start the real work!

Step 1: Divide the First Terms

Divide the first term of the dividend, $25y^3$, by the first term of the divisor, $5y$. That's $25y^3 / 5y = 5y^2$. This $5y^2$ is the first term of our quotient. So, we write $5y^2$ above the $y^2$ term in the dividend. This is where we begin to build our answer.

Step 2: Multiply

Next, multiply the entire divisor, $5y - 2$, by $5y^2$. This gives us $5y^2 * (5y - 2) = 25y^3 - 10y^2$. We write this result below the dividend, aligning like terms. It’s crucial to align the like terms correctly to prevent errors in the subtraction step.

Step 3: Subtract

Now, subtract the result we just obtained, $25y^3 - 10y^2$, from the corresponding terms in the dividend, $25y^3 - 25y^2$. This looks like $(25y^3 - 25y^2) - (25y^3 - 10y^2) = -15y^2$. So, we have $-15y^2$ left after the subtraction. Subtraction is the trickiest part for most people, so double-check your signs.

Step 4: Bring Down the Next Term

Bring down the next term from the original dividend, which is $-9y$. We add this to our result from the subtraction, giving us $-15y^2 - 9y$. Bringing down the terms correctly is vital for continuing the division process.

Step 5: Repeat

Now we repeat the process. Divide the first term of our new polynomial, $-15y^2$, by the first term of the divisor, $5y$. That's $-15y^2 / 5y = -3y$. This is the next term of our quotient, so we write $-3y$ next to the $5y^2$ in the quotient. Keep building the quotient step by step.

Multiply the divisor, $5y - 2$, by $-3y$. This gives us $-3y * (5y - 2) = -15y^2 + 6y$. Write this below $-15y^2 - 9y$, aligning like terms. Again, alignment is key.

Subtract this result from $-15y^2 - 9y$: $(-15y^2 - 9y) - (-15y^2 + 6y) = -15y$. Watch out for those negative signs! It’s easy to make a mistake here.

Bring down the next term, which is $-29$. Our new polynomial is $-15y - 29$. Almost there, guys! We're on the final stretch.

Repeat the process one last time. Divide the first term, $-15y$, by the first term of the divisor, $5y$. That's $-15y / 5y = -3$. Write $-3$ next to the other terms in the quotient. The quotient is really taking shape now.

Multiply the divisor, $5y - 2$, by $-3$. This gives us $-3 * (5y - 2) = -15y + 6$. Write this below $-15y - 29$, aligning like terms. Consistency is the name of the game.

Subtract this result from $-15y - 29$: $(-15y - 29) - (-15y + 6) = -35$. This is our remainder. The remainder is what's left over after the division.

The Quotient and Remainder

So, after all that, we've found that when we divide $25y^3 - 25y^2 - 9y - 29$ by $5y - 2$, the quotient is $5y^2 - 3y - 3$ and the remainder is $-35$. That’s the heart of the problem right there.

Determining if the Divisor is a Factor

Now, the final piece of the puzzle: Is $5y - 2$ a factor of $25y^3 - 25y^2 - 9y - 29$? Remember, a polynomial is a factor of another polynomial if the remainder is zero. This is a crucial concept.

The Remainder's Role

In our case, the remainder is $-35$, which is definitely not zero. So, $5y - 2$ is not a factor of $25y^3 - 25y^2 - 9y - 29$. If the remainder had been zero, then $5y - 2$ would have been a factor. The remainder tells the whole story.

Conclusion: No, It's Not a Factor

Therefore, the answer to the question "Is $5y - 2$ a factor of $25y^3 - 25y^2 - 9y - 29$?" is No. We've successfully completed the long division, found the quotient and remainder, and used the remainder to determine whether the divisor is a factor. We did it, guys!

Practice Makes Perfect: Tips and Tricks for Long Division

Polynomial long division can be a bit tricky at first, but with practice, you'll get the hang of it. Here are a few tips and tricks to help you master this skill. Let’s make sure you’re set up for success.

Stay Organized

One of the biggest challenges with long division is keeping everything organized. Make sure to align like terms vertically. Use plenty of space and write clearly. A messy setup can easily lead to errors. Organization is your best friend in math.

Double-Check Your Subtraction

Subtraction is where many mistakes happen, especially with negative signs. Take your time and double-check each subtraction step. It's helpful to rewrite the subtraction as addition by changing the signs of the terms being subtracted. Subtraction is a common stumbling block, so take extra care.

Practice, Practice, Practice

The best way to improve your long division skills is to practice. Work through plenty of problems, starting with simpler ones and gradually moving to more complex ones. The more you practice, the more comfortable you'll become with the process. There’s no substitute for practice! So, roll up your sleeves and get to it.

Use Online Resources

There are tons of great online resources that can help you with polynomial long division. Khan Academy, for example, has excellent videos and practice exercises. Symbolab and other math problem solvers can also help you check your work. Don't be afraid to use all the tools at your disposal.

Work Through Examples

When you're learning something new, it's often helpful to work through examples step by step. Find worked examples in your textbook or online and follow along carefully. Pay attention to each step and try to understand why it's being done. Examples are like a roadmap for solving problems.

Conclusion: Keep Practicing!

So, guys, we've covered a lot today! We've walked through polynomial long division step by step, solved a tricky problem, and learned how to determine if a divisor is a factor. Remember, math can be challenging, but with a solid understanding of the process and plenty of practice, you can master even the trickiest concepts. Keep up the great work, and you'll be a long division pro in no time!