Polynomial Division: Solving And Arranging In Descending Order

Polynomial division might sound intimidating at first, but fear not! It's a fundamental concept in algebra, and with a bit of practice, you'll master it. In this article, we'll dive deep into polynomial division, specifically focusing on the given expression: $\left(6 x^3+27 x-19 x^2-15\right) \div(3 x-5)=$. We'll break down the process step-by-step, ensuring you understand how to arrive at the correct answer. The key to success is understanding the logic behind each step. Let's start with a general overview of the process before we tackle our specific problem. Polynomial division is similar to long division with numbers, but instead of dividing numbers, we divide polynomials. The goal is to find a quotient and a remainder, just like in regular division. To start, make sure to write the polynomial in descending order of powers of the variable (in this case, x). Also, ensure that all terms are present; if a term is missing, we can add a zero coefficient to act as a placeholder. This will help you keep the terms organized and reduce the chance of errors. Furthermore, the divisor's form, $(3x-5)$ in our problem, plays an important role, influencing each step of the division. Understanding these basics is crucial to successfully implementing the division algorithm. Don't worry if it sounds a bit complicated now; we'll clear everything up as we go through the example.

Arranging and Setting Up the Division

Before we begin the division, let's rearrange the dividend (the polynomial being divided) in descending order of powers of x. Our dividend is $6x^3 + 27x - 19x^2 - 15$. Arranging this in descending order, we get $6x^3 - 19x^2 + 27x - 15$. Now, we can set up the long division problem. Write the dividend inside the division symbol and the divisor $(3x - 5)$ outside. Ensure the setup is clean and well-organized, as this is crucial for the correct execution of the division. Pay close attention to the terms: make sure to align the terms with the same powers of x in the dividend and the quotient. This is especially important as you progress through each iterative step of the process. Remember, we will be going through several cycles of division, multiplication, and subtraction to arrive at the final solution. Getting the initial setup right is extremely important, as it will affect all subsequent calculations. Remember, the goal is to systematically reduce the dividend until we are left with a remainder that is either zero or has a degree (power of x) less than that of the divisor. We are now ready to begin the division process and compute the quotient and remainder.

Step-by-Step Polynomial Division

Now, let's perform the polynomial division. Here's a detailed, step-by-step breakdown:

1. Divide the first term of the dividend by the first term of the divisor: Divide $6x^3$ (the first term of the dividend) by $3x$ (the first term of the divisor). This gives us $2x^2$. This is the first term of our quotient. Write $2x^2$ above the division symbol, aligning it with the $x^2$ term in the dividend.

2. Multiply the divisor by the result: Multiply the entire divisor $(3x - 5)$ by $2x^2$. This gives us $6x^3 - 10x^2$. Write this result below the dividend, aligning the terms with their corresponding powers of x.

3. Subtract and bring down the next term: Subtract the result obtained in step 2 from the dividend. $(6x^3 - 19x^2) - (6x^3 - 10x^2) = -9x^2$. Bring down the next term from the dividend, which is $+27x$. Now we have $-9x^2 + 27x$.

4. Repeat the process: Divide the first term of the new expression, $-9x^2$, by the first term of the divisor, $3x$. This gives us $-3x$. Write $-3x$ as the second term of the quotient.

5. Multiply: Multiply the divisor $(3x - 5)$ by $-3x$. This gives us $-9x^2 + 15x$. Write this below $-9x^2 + 27x$.

6. Subtract and bring down: Subtract $(-9x^2 + 15x)$ from $(-9x^2 + 27x)$. This leaves us with $12x$. Bring down the next term from the dividend, which is $-15$. Now we have $12x - 15$.

7. Final Repeat: Divide $12x$ by $3x$, which results in $4$. Write $+4$ as the next term in the quotient.

8. Multiply: Multiply the divisor $(3x - 5)$ by $4$. This gives us $12x - 20$. Write this below $12x - 15$.

9. Subtract to Find the Remainder: Subtract $(12x - 20)$ from $(12x - 15)$. This results in a remainder of $5$.

Therefore, we have a quotient of $2x^2 - 3x + 4$ and a remainder of $5$. Therefore, the result of the division is $2x^2 - 3x + 4 + \frac{5}{3x - 5}$.

Matching with the Answer Choices

Now that we've performed the polynomial division and found our quotient and remainder, let's see how this matches up with the given answer choices. We found that the result of the division is $2x^2 - 3x + 4 + \frac{5}{3x - 5}$. Examining the provided answer choices:

• A. $4 x^2-3 x+2+\frac{3}{5 x-5}$
• B. $2 x^2-3 x+4+\frac{5}{3 x-5}$
• C. $2 x^2-3 x+4+\frac{5}{3 x-5}$

We can see that the correct answer is C. This is because it matches exactly with our calculated quotient and remainder. It's crucial to ensure every term in the quotient and remainder, as well as the format of the remainder (the fraction), aligns perfectly with the answer options provided. In this instance, the correct selection is evident once you have correctly completed the polynomial division. Remembering to express the final answer in the correct format, with the remainder term, is also important. The remainder must be correctly placed over the original divisor.

Conclusion: Mastering Polynomial Division

By following these steps, you can successfully perform polynomial division and find the quotient and remainder. Always remember to arrange the dividend in descending order of powers, divide the leading terms, multiply, subtract, and bring down the next term, repeat the process until the remainder's degree is less than the divisor's. In our example, the degree of our remainder (5) is 0 and is less than the degree of the divisor $(3x-5)$, which is 1. Practicing these problems will increase your understanding and make polynomial division a breeze.

We successfully performed the polynomial division for the given expression and identified the correct answer from the provided options. Understanding and correctly applying the steps of the polynomial division process is essential for success.

For further practice and more examples on this topic, check out these trusted resources:

• Khan Academy (https://www.khanacademy.org/) - Khan Academy offers free, comprehensive lessons on polynomial division and other algebraic concepts. It's a great platform to reinforce your understanding and practice with different examples.