Solving (f/g)(x) Polynomial Division Explained

Hey guys! Today, we're diving into a fun little math problem that involves dividing polynomials. Specifically, we're tasked with finding $(f/g)(x)$ given that $f(x) = x^4 - x^3 + x^2$ and $g(x) = -x^2$, where $x \neq 0$. Don't worry, it's not as scary as it looks! We'll break it down step by step.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. When we see $(f/g)(x)$, it simply means we need to divide the function $f(x)$ by the function $g(x)$. In other words, we need to calculate $\frac{f(x)}{g(x)}$. This is a classic example of polynomial division, a fundamental concept in algebra. You might remember polynomial division from your high school math classes, and it's a skill that comes in handy in various areas of mathematics and engineering. The key here is to handle the algebraic expressions carefully and simplify the result as much as possible. Remember, the condition $x \neq 0$ is crucial because it prevents division by zero, which is undefined in mathematics. So, let's keep that in mind as we proceed with the calculation.

The problem provides us with two polynomial functions:

• $f(x) = x^4 - x^3 + x^2$
• $g(x) = -x^2$

Our mission, should we choose to accept it (and we do!), is to find the result of dividing $f(x)$ by $g(x)$. We need to perform the division $\frac{x^4 - x^3 + x^2}{-x^2}$ and simplify the resulting expression. This involves dividing each term of the polynomial $f(x)$ by $g(x)$ and then combining the results. Think of it like distributing the division across each term. We will pay close attention to the signs and exponents to ensure we get the correct answer. The goal is to simplify the expression into one of the answer choices provided: A, B, C, or D. Each of these options represents a different quadratic polynomial, and our job is to manipulate the division until we arrive at one of them. So, let's get started and see how we can simplify this expression!

Step-by-Step Solution

Alright, let's tackle this polynomial division head-on! We're essentially trying to simplify the fraction:

$\frac{x^4 - x^3 + x^2}{-x^2}$

The best way to approach this is to divide each term in the numerator ($x^4 - x^3 + x^2$) by the denominator ($-x^2$). This is like splitting the fraction into three smaller fractions:

$\frac{x^4}{-x^2} - \frac{x^3}{-x^2} + \frac{x^2}{-x^2}$

Now, let's simplify each of these fractions individually. Remember the rules of exponents: when you divide terms with the same base, you subtract the exponents. Also, pay close attention to the signs!

• $\frac{x^4}{-x^2} = -x^{4-2} = -x^2$
• $\frac{-x^3}{-x^2} = x^{3-2} = x$
• $\frac{x^2}{-x^2} = -1$

So, putting it all together, we have:

$-x^2 + x - 1$

And there you have it! The simplified expression for $(f/g)(x)$ is $-x^2 + x - 1$. This matches option C in the given choices. This step-by-step process highlights the importance of breaking down complex problems into smaller, manageable parts. By focusing on each term individually, we were able to simplify the expression and arrive at the correct solution. It's a great strategy for tackling any math problem, especially those involving algebraic manipulation.

Choosing the Correct Option

After performing the division and simplifying the expression, we arrived at the result: $-x^2 + x - 1$. Now, let's compare this to the answer options provided:

A. $x^2 - x + 1$ B. $x^2 + x + 1$ C. $-x^2 + x - 1$ D. $-x^2 - x - 1$

By carefully comparing our result with the options, we can clearly see that it matches option C. The expression $-x^2 + x - 1$ is exactly what we calculated. Options A, B, and D have different signs and terms, so they are incorrect. Therefore, the correct answer is C. This final step of comparing the result with the given options is crucial in any problem-solving scenario. It ensures that we have not made any errors in our calculations and that we have selected the appropriate answer. Always double-check your work and compare your final answer with the available choices to avoid careless mistakes. This simple step can make the difference between a correct answer and a missed point.

Common Mistakes to Avoid

When dealing with polynomial division, there are a few common pitfalls that students often encounter. Let's highlight these mistakes so you can avoid them in the future:

1. Incorrectly Applying the Exponent Rules: Remember that when dividing terms with exponents, you subtract the exponents, not divide them. For example, $\frac{x^4}{x^2}$ is $x^{4-2} = x^2$, not $x^{4/2} = x^2$ (which happens to be the same in this case, but the process is different!). Always double-check your exponent arithmetic.
2. Sign Errors: Pay very close attention to the signs, especially when dealing with negative terms. A simple sign error can throw off the entire calculation. For instance, dividing a positive term by a negative term will result in a negative term. Be meticulous and double-check your signs at each step. Sign errors are a very common source of mistakes in algebra, so be vigilant!
3. Forgetting to Divide All Terms: When dividing a polynomial by a monomial (a single term), you need to divide each term in the polynomial by the monomial. Don't forget to distribute the division across all terms. It's easy to overlook a term, so make sure you've accounted for everything.
4. Not Simplifying Completely: After performing the division, make sure you simplify the expression as much as possible. Combine like terms and reduce fractions to their simplest form. Leaving an expression unsimplified can lead to an incorrect answer if it doesn't match the given options.
5. Ignoring the Restriction on x: In this problem, we had the condition $x \neq 0$. While it didn't directly affect the calculation in this case, it's important to acknowledge this restriction. Ignoring such conditions can lead to misunderstandings in more complex problems.

By being aware of these common mistakes, you can increase your accuracy and confidence when solving polynomial division problems. Remember, practice makes perfect, so keep working on these types of problems to solidify your understanding.

Practice Makes Perfect

To really master polynomial division, practice is key! Here are a few similar problems you can try to solidify your understanding:

1. If $f(x) = 2x^3 + 4x^2 - 6x$ and $g(x) = 2x$, what is $(f/g)(x)$?
2. If $f(x) = x^5 - 3x^3 + x$ and $g(x) = x$, what is $(f/g)(x)$?
3. If $f(x) = -4x^4 + 8x^2$ and $g(x) = -2x^2$, what is $(f/g)(x)$?

Working through these problems will help you become more comfortable with the process of polynomial division and avoid common mistakes. Remember to break down the problems into smaller steps, pay attention to signs and exponents, and simplify your answers completely. And most importantly, have fun with it! Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem. So, grab a pencil and paper, and get practicing! The more you practice, the more confident you'll become in your math skills. Good luck, and happy problem-solving!

Conclusion

In conclusion, we've successfully navigated the world of polynomial division to find $(f/g)(x)$ for the given functions $f(x)$ and $g(x)$. By breaking down the problem into manageable steps, carefully applying the rules of exponents and signs, and avoiding common mistakes, we arrived at the correct answer: $-x^2 + x - 1$. This corresponds to option C in the provided choices. This exercise highlights the importance of a systematic approach to problem-solving, particularly in algebra. Remember to always read the problem carefully, understand what's being asked, and break down complex tasks into smaller, more manageable steps. Pay attention to detail, double-check your work, and practice regularly to build your skills and confidence.

Polynomial division is a fundamental concept in mathematics, and mastering it will serve you well in more advanced topics. So, keep practicing, keep learning, and don't be afraid to tackle challenging problems. With a little effort and the right approach, you can conquer any mathematical obstacle. Remember, every problem you solve is a step forward in your mathematical journey. So, keep stepping forward and enjoy the ride!