(2x^3+3x^2+4x+5)/(2x^2+3x+4)

3 min read Jun 16, 2024

Performing Polynomial Long Division: (2x^3 + 3x^2 + 4x + 5) / (2x^2 + 3x + 4)

In this article, we will walk through the process of dividing the polynomial 2x^3 + 3x^2 + 4x + 5 by 2x^2 + 3x + 4 using polynomial long division.

1. Setting up the Division

We start by setting up the division problem in the same way we would with numerical long division.

             ________
2x^2+3x+4 | 2x^3 + 3x^2 + 4x + 5

2. Dividing the Leading Terms

We focus on the leading terms of both the divisor and the dividend.

2x^3 (leading term of the dividend) divided by 2x^2 (leading term of the divisor) equals x.
We write x above the line, aligning it with the x^2 term in the dividend.

             x ______
2x^2+3x+4 | 2x^3 + 3x^2 + 4x + 5

3. Multiplying and Subtracting

Multiply the divisor (2x^2 + 3x + 4) by x and write the result below the dividend.
Subtract the resulting expression from the dividend.

             x ______
2x^2+3x+4 | 2x^3 + 3x^2 + 4x + 5 
             -(2x^3 + 3x^2 + 4x)
             -------------------
                        0 + 5

4. Bringing Down the Next Term

Bring down the next term of the dividend (in this case, + 5)

             x ______
2x^2+3x+4 | 2x^3 + 3x^2 + 4x + 5 
             -(2x^3 + 3x^2 + 4x)
             -------------------
                        0 + 5

5. Repeating the Process

Now, the new dividend is 5. Since the degree of 5 (degree 0) is less than the degree of the divisor (degree 2), we cannot continue the division.

6. Result

The result of the division is:

Quotient: x
Remainder: 5

Therefore, we can write the expression as:

(2x^3 + 3x^2 + 4x + 5) / (2x^2 + 3x + 4) = x + 5 / (2x^2 + 3x + 4)

This result means that the original polynomial (2x^3 + 3x^2 + 4x + 5) can be expressed as x times the divisor (2x^2 + 3x + 4) plus the remainder 5.