Polynomial Division Explained Find Remainder And Quotient Of 2n^3-7n^2+n+10

Jul 10, 2025 by Admin 76 views

Unveiling the Remainder and Quotient: A Deep Dive into Polynomial Division

In the fascinating world of mathematics, polynomial division stands as a cornerstone, enabling us to dissect complex expressions and unearth their underlying structure. This exploration delves into a specific instance of polynomial division, offering a comprehensive guide on determining the remainder and quotient when dividing the polynomial $2n^3 - 7n^2 + n + 10$ by $(n - 1)$ . We will use synthetic division method and polynomial long division method.

The Essence of Polynomial Division

Polynomial division, at its core, mirrors the familiar process of numerical long division. It involves dividing a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder. The fundamental relationship governing this process is expressed as:

Dividend = (Divisor × Quotient) + Remainder

In our case, the dividend is $2n^3 - 7n^2 + n + 10$ , the divisor is $(n - 1)$ , and our quest is to find the quotient, denoted as $Q(n)$ , and the remainder, represented by $R$ .

Method 1: Synthetic Division

Synthetic division is a streamlined technique for dividing a polynomial by a linear divisor of the form $(n - c)$ . It offers a concise and efficient way to determine both the quotient and the remainder. To embark on synthetic division, we set up a table with the coefficients of the dividend and the value of $c$ from the divisor. For our problem, the coefficients are 2, -7, 1, and 10, and $c$ is 1.

The synthetic division process unfolds as follows:

Write down the coefficients of the polynomial and the root of the divisor.
```
1 | 2 -7 1 10
```
Bring down the leading coefficient.
```
1 | 2 -7 1 10
| 2
----------------
2
```
Multiply the leading coefficient by the root and write the result under the next coefficient.
```
1 | 2 -7 1 10
| 2
----------------
2
```

Add the numbers in the second column.

1 | 2 -7 1 10
| 2
----------------
2 -5

Repeat steps 3 and 4 for the remaining columns.

1 | 2 -7 1 10
| 2 -5 -4
----------------
2 -5 -4 6

The last number in the bottom row is the remainder, and the other numbers are the coefficients of the quotient.

From the synthetic division, we discern that the quotient $Q(n)$ is $2n^2 - 5n - 4$ , and the remainder $R$ is 6.

Method 2: Polynomial Long Division

Polynomial long division mirrors the familiar numerical long division algorithm, providing a systematic approach to dividing polynomials. While it may appear more verbose than synthetic division, it offers a versatile technique applicable to divisors of any degree. The steps involved in polynomial long division are as follows:

Set up the division: Arrange the dividend ( $2n^3 - 7n^2 + n + 10$ ) and the divisor ( $n - 1$ ) in a long division format.

 n - 1 | 2n^3 - 7n^2 + n + 10

Divide the leading terms: Divide the leading term of the dividend ( $2n^3$ ) by the leading term of the divisor ( $n$ ) to obtain the first term of the quotient ( $2n^2$ ).

 2n^2
n - 1 | 2n^3 - 7n^2 + n + 10

Multiply the divisor by the quotient term: Multiply the divisor ( $n - 1$ ) by the first term of the quotient ( $2n^2$ ) to get $2n^3 - 2n^2$ .

 2n^2
n - 1 | 2n^3 - 7n^2 + n + 10
 2n^3 - 2n^2

Subtract and bring down: Subtract the result from the corresponding terms of the dividend and bring down the next term ( $+n$ ).

 2n^2
n - 1 | 2n^3 - 7n^2 + n + 10
 -(2n^3 - 2n^2)
 -----------------
 -5n^2 + n

Repeat the process: Repeat steps 2-4 with the new polynomial ( $-5n^2 + n$ ). Divide $-5n^2$ by $n$ to get $-5n$ , multiply $(n - 1)$ by $-5n$ to get $-5n^2 + 5n$ , subtract, and bring down the next term (+10).

 2n^2 - 5n
n - 1 | 2n^3 - 7n^2 + n + 10
 -(2n^3 - 2n^2)
 -----------------
 -5n^2 + n
 -(-5n^2 + 5n)
 -----------------
 -4n + 10

Continue until the degree of the remainder is less than the degree of the divisor: Repeat the process one more time. Divide $-4n$ by $n$ to get $-4$ , multiply $(n - 1)$ by $-4$ to get $-4n + 4$ , and subtract.

 2n^2 - 5n - 4
n - 1 | 2n^3 - 7n^2 + n + 10
 -(2n^3 - 2n^2)
 -----------------
 -5n^2 + n
 -(-5n^2 + 5n)
 -----------------
 -4n + 10
 -(-4n + 4)
 --------------
 6

Identify the quotient and remainder: The quotient is the polynomial at the top ( $2n^2 - 5n - 4$ ), and the remainder is the final value at the bottom (6).

The result of polynomial long division corroborates our findings from synthetic division: the quotient $Q(n)$ is $2n^2 - 5n - 4$ , and the remainder $R$ is 6.

Verification

To ensure the accuracy of our results, we can substitute the obtained quotient and remainder back into the fundamental relationship:

Dividend = (Divisor × Quotient) + Remainder

$2n^3 - 7n^2 + n + 10 = (n - 1)(2n^2 - 5n - 4) + 6$

Expanding the right-hand side, we get:

$2n^3 - 5n^2 - 4n - 2n^2 + 5n + 4 + 6$

Simplifying, we arrive at:

$2n^3 - 7n^2 + n + 10$

This confirms that our calculated quotient and remainder are indeed correct.

Conclusion

Through the application of both synthetic division and polynomial long division, we have successfully determined the quotient and remainder when dividing $2n^3 - 7n^2 + n + 10$ by $(n - 1)$ . The quotient $Q(n)$ is $2n^2 - 5n - 4$ , and the remainder $R$ is 6. This exercise underscores the power and elegance of polynomial division in unraveling the structure of algebraic expressions. Understanding polynomial division is crucial for various mathematical applications, such as solving equations, factoring polynomials, and simplifying rational expressions. By mastering these techniques, students can gain a deeper appreciation for the interconnectedness of mathematical concepts and enhance their problem-solving abilities. Remember, the journey of mathematical exploration is a continuous one, and each problem solved is a step forward in expanding our understanding of the world around us. So, embrace the challenge, delve into the intricacies of polynomial division, and unlock the beauty of mathematics!

Polynomial division is not just a mathematical exercise; it's a gateway to understanding more complex concepts in algebra and calculus. For example, the Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - c$ , then the remainder is $f(c)$ . In our case, if we substitute $n = 1$ into the original polynomial $2n^3 - 7n^2 + n + 10$ , we get $2(1)^3 - 7(1)^2 + 1 + 10 = 2 - 7 + 1 + 10 = 6$ , which is indeed the remainder we found. This theorem provides a quick way to find the remainder without performing the full division process, and it reinforces the connection between division and function evaluation.

Furthermore, polynomial division is closely related to factoring polynomials. If the remainder is zero, it means that the divisor is a factor of the dividend. This is the essence of the Factor Theorem, which is a cornerstone of polynomial factorization. In our example, since the remainder is 6 (not zero), $(n - 1)$ is not a factor of $2n^3 - 7n^2 + n + 10$ . However, if we had a remainder of zero, we could confidently say that the divisor is a factor, and we could proceed to find other factors and ultimately solve polynomial equations.

The quotient we obtained, $2n^2 - 5n - 4$ , is a quadratic polynomial. We can further investigate whether this quadratic can be factored. Factoring quadratics is a fundamental skill in algebra, and it often involves techniques like finding two numbers that multiply to the constant term and add up to the coefficient of the linear term. In this case, we can try to factor $2n^2 - 5n - 4$ , but it doesn't factor neatly into integers. This means that the roots of this quadratic are not rational, and we would need to use the quadratic formula to find them.

In conclusion, polynomial division is a powerful tool that unlocks a deeper understanding of polynomial structure and behavior. It's not just about finding quotients and remainders; it's about connecting concepts, verifying theorems, and laying the groundwork for more advanced mathematical techniques. By mastering polynomial division, students can build a strong foundation for future studies in mathematics and related fields. So, continue practicing, continue exploring, and continue to marvel at the beauty and interconnectedness of mathematical ideas!