Synthetic Division Find Quotient And Remainder Of X^4-9x^2+3x+5 Divided By X-3

Hey guys! Let's dive into the wonderful world of synthetic division! It's a super handy tool in mathematics, especially when we need to divide polynomials. Today, we're going to break down how to use synthetic division to find the quotient and remainder when we divide a polynomial by a linear expression. Specifically, we'll tackle the example of dividing $x^4 - 9x^2 + 3x + 5$ by $x - 3$. So, grab your pencils, and let's get started!

Understanding Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear expression of the form $x - c$. It's way quicker than long division, especially for higher-degree polynomials. Think of it as a shortcut that helps us find the quotient and remainder without all the messy steps of long division. The key is to focus on the coefficients of the polynomial and the constant term of the divisor. This method is particularly useful in various algebraic manipulations, such as finding roots of polynomials and factoring. Before we jump into our example, let's quickly review the basic process of synthetic division. First, we identify the coefficients of the polynomial and write them down. Remember, if a term is missing (like the $x^3$ term in our example), we need to include a zero as a placeholder. Next, we identify the value of c from the divisor x - c. In our case, the divisor is x - 3, so c is 3. We then set up our synthetic division table and follow a series of steps involving bringing down coefficients, multiplying, and adding. The final row of the table gives us the coefficients of the quotient and the remainder. With synthetic division, complex polynomial divisions become manageable and less prone to errors. It’s a powerful tool that simplifies the process and saves time, making it an essential technique for anyone working with polynomials.

Setting Up the Problem: Dividing $x^4 - 9x^2 + 3x + 5$ by $x - 3$

To kick things off, we need to set up our synthetic division problem correctly. The polynomial we're dividing is $x^4 - 9x^2 + 3x + 5$, and we're dividing it by $x - 3$. First, let's identify the coefficients of our polynomial. We have a term for $x^4$, but notice we're missing an $x^3$ term. We can think of our polynomial as $1x^4 + 0x^3 - 9x^2 + 3x + 5$. So, our coefficients are 1, 0, -9, 3, and 5. It's crucial to include that 0 for the missing $x^3$ term! For the divisor $x - 3$, we take the value of c, which is 3. This is the number we'll use in our synthetic division process. Now, we set up our synthetic division table. We write the value of c (which is 3) to the left. Then, we write the coefficients of our polynomial (1, 0, -9, 3, and 5) in a row to the right. We draw a horizontal line under the coefficients, leaving space below for our calculations. This setup is the foundation for the synthetic division process. It's essential to get this right, as it sets the stage for the rest of the calculation. Ensuring that all coefficients are correctly identified and placed, including any necessary placeholders for missing terms, will lead to an accurate result. The setup might seem simple, but it's the cornerstone of successful synthetic division. By taking our time and paying attention to detail in this initial step, we ensure that the subsequent calculations are based on a solid foundation, leading to the correct quotient and remainder.

Performing Synthetic Division: Step-by-Step

Now, let’s get into the heart of the synthetic division process. Remember, we're dividing $x^4 - 9x^2 + 3x + 5$ by $x - 3$. We've already set up our table with the coefficients (1, 0, -9, 3, 5) and the value 3 from the divisor. The first step is to bring down the first coefficient, which is 1, below the horizontal line. This 1 will be the leading coefficient of our quotient. Next, we multiply this 1 by the value we're dividing by, which is 3. So, 1 times 3 equals 3. We write this 3 under the next coefficient, which is 0. Now, we add the numbers in this column: 0 plus 3 equals 3. We write this 3 below the line. We repeat the process: multiply 3 by 3 to get 9, and write it under the next coefficient, -9. Add -9 and 9 to get 0. Write the 0 below the line. Multiply 0 by 3 to get 0, and write it under the next coefficient, 3. Add 3 and 0 to get 3. Write the 3 below the line. Finally, multiply 3 by 3 to get 9, and write it under the last coefficient, 5. Add 5 and 9 to get 14. This final number, 14, is our remainder. The numbers below the line (1, 3, 0, 3) are the coefficients of our quotient. Synthetic division might seem like a series of steps, but with practice, it becomes second nature. Each step builds upon the previous one, leading us closer to the solution. The process of multiplying and adding ensures that we're systematically accounting for each term in the polynomial division. By carefully following these steps, we can efficiently and accurately find the quotient and remainder, making synthetic division a valuable tool in polynomial algebra.

Interpreting the Results: Quotient and Remainder

Alright, we've gone through the synthetic division process, and now it's time to interpret our results. Remember, we divided $x^4 - 9x^2 + 3x + 5$ by $x - 3$, and our synthetic division gave us the numbers 1, 3, 0, 3, and 14 below the line. These numbers are the key to finding our quotient and remainder. The last number, 14, is the remainder. So, $R(x) = 14$. The other numbers (1, 3, 0, 3) are the coefficients of the quotient. Since we started with an $x^4$ polynomial and divided by $x$, our quotient will be a polynomial of degree one less, which is $x^3$. The coefficients 1, 3, 0, and 3 correspond to the terms $x^3$, $x^2$, $x$, and the constant term, respectively. So, our quotient is $1x^3 + 3x^2 + 0x + 3$, which simplifies to $x^3 + 3x^2 + 3$. Therefore, $Q(x) = x^3 + 3x^2 + 3$. In summary, when we divide $x^4 - 9x^2 + 3x + 5$ by $x - 3$, the quotient is $x^3 + 3x^2 + 3$, and the remainder is 14. Understanding how to interpret these results is just as important as performing the synthetic division itself. The coefficients we obtain directly translate into the terms of the quotient, and the final number gives us the remainder. By correctly interpreting these values, we can confidently state the result of our polynomial division. This skill is crucial for various applications in algebra, such as factoring polynomials, finding roots, and solving equations.

Final Answer and Conclusion

So, to wrap it all up, we've successfully used synthetic division to find the quotient and remainder when $x^4 - 9x^2 + 3x + 5$ is divided by $x - 3$. We found that the quotient, $Q(x)$, is $x^3 + 3x^2 + 3$, and the remainder, $R(x)$, is 14. Awesome job, guys! Synthetic division is a powerful tool that simplifies polynomial division, and with a bit of practice, you can master it too. Remember, the key is to set up the problem correctly, follow the steps carefully, and interpret the results accurately. Whether you're working on homework, studying for a test, or just exploring the world of algebra, synthetic division is a skill that will definitely come in handy. By understanding and applying this method, you can tackle polynomial division problems with confidence and efficiency. So keep practicing, and you'll become a synthetic division pro in no time! Keep up the great work! And don't forget, mathematics is all about practice, so the more you do it, the better you'll get. You've now added a valuable tool to your mathematical toolkit, one that will help you in countless problems to come. Keep exploring, keep learning, and most importantly, keep having fun with math! You've got this!