Finding The Remainder: Dividing Polynomials With Precision

Hey guys! Let's dive into some cool math stuff! We're gonna explore polynomial division and figure out remainders. The problem gives us two functions: $g(x) = 3x^2 - 2x$ and $f(x) = 6x^4 + 5x^3 + 3x - 5$. The question is, what happens when we divide $f(x)$ by $g(x)$? Specifically, we want to know about the remainder! This is super useful in algebra. Understanding remainders helps us see if one polynomial is a factor of another and unravel other exciting mathematical puzzles. It's like a detective game, and we are the math detectives! The heart of this lies in polynomial division, a technique to systematically break down a larger polynomial by a smaller one, unveiling the remainder and quotient along the way. Stay with me, because we will break down the process step by step, making it easy to understand. We'll find out if the remainder is zero, if g(x) is a factor of f(x) or not. It might sound complicated, but it's really about organization and being methodical. Let's get started!

Decoding the Remainder: A Step-by-Step Guide

Alright, let's break this down piece by piece. When we divide one polynomial by another, the remainder is what's left over after the division is complete. If the remainder is zero, it means the divisor goes into the dividend perfectly, like a clean cut. In other words, the divisor is a factor of the dividend. But if there's a non-zero remainder, then the divisor doesn't divide the dividend evenly. Now, let's get down to the actual division. We'll be using polynomial long division.

So first, we will set up the division problem. The dividend ($f(x) = 6x^4 + 5x^3 + 3x - 5$) goes inside the division symbol, and the divisor ($g(x) = 3x^2 - 2x$) goes outside. Remember to include any missing terms with a coefficient of zero for neatness. This helps keep things organized. In this case, there are no missing terms, which makes things easier! Let’s perform the first step. We look at the leading terms of both the dividend ($6x^4$) and the divisor ($3x^2$). We divide the leading term of the dividend by the leading term of the divisor: $(6x^4) / (3x^2) = 2x^2$. This is the first term of our quotient. Now, we multiply the entire divisor ($3x^2 - 2x$) by this first term of the quotient ($2x^2$). This gives us $2x^2 * (3x^2 - 2x) = 6x^4 - 4x^3$. Write this result under the dividend, aligning like terms. Now, subtract this result from the dividend. This is a crucial step! Subtracting the polynomials gives us: $(6x^4 + 5x^3 + 3x - 5) - (6x^4 - 4x^3) = 9x^3 + 3x - 5$. Next, we bring down any remaining terms from the dividend. Since there's nothing left to bring down, we are done with this step.

Continuing the Division: Getting Closer to the Solution

Now, we repeat the process. We look at the leading term of the new polynomial ($9x^3$) and divide it by the leading term of the divisor ($3x^2$). So, $(9x^3) / (3x^2) = 3x$. This becomes the next term in our quotient. We then multiply the divisor ($3x^2 - 2x$) by this new term ($3x$): $3x * (3x^2 - 2x) = 9x^3 - 6x^2$. Write this result under the current polynomial and align like terms. Subtract again: $(9x^3 + 3x - 5) - (9x^3 - 6x^2) = 6x^2 + 3x - 5$. Bring down any remaining terms; in this case, there are none.

We perform another round of division. Now, we look at the leading term of the current polynomial ($6x^2$) and divide it by the leading term of the divisor ($3x^2$). So, $(6x^2) / (3x^2) = 2$. This is the next term in our quotient. Multiply the divisor ($3x^2 - 2x$) by this new term ($2$): $2 * (3x^2 - 2x) = 6x^2 - 4x$. Write this under the current polynomial and align like terms. Subtract: $(6x^2 + 3x - 5) - (6x^2 - 4x) = 7x - 5$. And this is our remainder, since the degree of $7x - 5$ is less than the degree of the divisor $3x^2 - 2x$. We can't divide it further. Therefore, the remainder is $7x - 5$. So cool, right?

Unveiling the Final Answer and Its Significance

Okay, folks, we're at the finish line! After all that division, we've found our remainder. It's $7x - 5$. Now, let's look at the options. We can immediately eliminate the first option, (A), because the remainder isn't zero. That means $g(x)$ is not a factor of $f(x)$. The other options would give us information on what the remainder is. In our case, the remainder is $7x - 5$. The remainder isn't zero, which means that $g(x)$ is not a factor of $f(x)$. So the final answer is $7x-5$ which is not zero.

So, what does this all mean? The remainder tells us how much 'extra' is left over after we divide $f(x)$ by $g(x)$. Since the remainder is not zero, $g(x)$ does not divide $f(x)$ perfectly. Polynomial division is a cornerstone of algebra, allowing us to simplify complex expressions, find roots of equations, and explore the relationships between different polynomials. This ability to break down polynomials is key to solving a wide variety of mathematical problems! It also helps in graphing, and understanding the behavior of functions. The ability to find the remainder gives us more insight into how these functions behave.

Recap and Key Takeaways

Alright, let's do a quick recap. We used polynomial long division to divide $f(x)$ by $g(x)$. We systematically divided the leading terms, multiplied, subtracted, and brought down terms until we got a remainder. The remainder was $7x - 5$. This process highlights the power of structured thinking. Breaking a problem down into smaller, manageable steps makes even complex math problems accessible. Understanding remainders helps us understand factors, roots, and the overall behavior of polynomials. So, the next time you see a polynomial division problem, remember the steps, stay organized, and you'll find the remainder like a pro!

This whole process might seem a bit long, but trust me, with practice, it becomes easier. Keep practicing those problems, and you will be amazing. Keep up the amazing work! You guys got this!