how to divide polynomials using long division

Polynomial long division divides a polynomial by another polynomial of the same or lower degree by repeating a four-step cycle: Divide, Multiply, Subtract, and Bring Down. It works just like regular long division with numbers. [1, 2, 3]

Step-by-Step Method

1. Set up the problem. Write both polynomials in descending order of exponents. Place the dividend (numerator) inside the division bracket and the divisor (denominator) on the outside left. If any power of x is missing, insert a placeholder with a coefficient of 0 (such as 0x²) so columns stay aligned. [4, 5]
2. Divide the leading terms. Divide the first term of the dividend by the first term of the divisor. Write the result on top of the division bracket. [2, 4]
3. Multiply the divisor. Multiply the term you just wrote on top by every term in the divisor. Write this product directly beneath the matching terms inside the bracket. [1, 4]
4. Subtract the bottom row. Subtract the entire product from the line above it. To avoid sign errors, flip the plus or minus sign of every term in the bottom row and add downward. [1, 4]
5. Bring down and repeat. Bring down the next term from the dividend. Repeat steps 2 through 4 until the degree of the remaining expression is smaller than the degree of the divisor.
6. Write the final answer. Combine the terms on top as your quotient. If a remainder is left over, write it as a fraction over the original divisor and add it to the end. [1, 5]

Worked Example

Divide (2x³ + x² – 6x – 8) by (x – 2):

2x² + 5x + 4

x – 2 | 2x³ + x² – 6x – 8

-(2x³ – 4x²)

5x² – 6x

-(5x² – 10x)

4x – 8

-(4x – 8)

0

*

• Cycle 1. Divide 2x³ by x to get 2x². Multiply 2x²(x – 2) to get 2x³ – 4x², subtract to get 5x², and bring down -6x.
• Cycle 2. Divide 5x² by x to get +5x. Multiply 5x(x – 2) to get 5x² – 10x, subtract to get 4x, and bring down -8.
• Cycle 3. Divide 4x by x to get +4. Multiply 4(x – 2) to get 4x – 8, and subtract to get a remainder of 0.
• Result. The final quotient is 2x² + 5x + 4. (If the remainder had been 3, you would write $2x^2 + 5x + 4 + \frac{3}{x – 2}$). [2, 6]

*

For extra practice problems and visual diagrams, see or.

If you have a specific polynomial division problem you are working on, share it here and I can walk you through the exact steps and final answer.