# Polynomial division

This calculator divides one polynomial by another polynomial.

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Created: 2018-02-15 10:20:04, Last updated: 2021-02-17 10:28:29

This calculator divides a higher degree polynomial by a lower degree polynomial. As a result, it gives a polynomial quotient and remainder. The polynomial division algorithm is explained just after the calculator:

#### Polynomial division

The polynomial coefficients, space separated, in order from higher term degree to lower
Divider polynomial coefficients, space separated, in order from higher term degree to lower
Dividend

Divisor

Result

Remainder

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### Polynomial division step by step

• Write down dividend polynomial in a row, including zero terms.
• Determine the first result term by dividing the highest degree dividend polynomial term by the highest degree divisor term.
• Multiply the divisor polynomial by the previous step result.
• Write down the previous step result just below the original polynomial. Same degree terms are one under the other
• Subtract the polynomial obtained on the previous step from the original polynomial or previous remainder polynomial.
• Write down the remainder on the next row skipping all leading terms turned to zero.
• Repeat all the steps above except the first one if the remainder polynomial degree is higher or equal to the divisor degree.
• Otherwise (if the remainder polynomial degree is lower than the divisor degree), the division is completed. The sum of terms obtained in step 2 is the quotient polynomial.

Let's consider division example: 3x4+5x3+2x+4 / x2+2x+1.

x4 x3 x2 x x0 Description Result terms
+3x4
+3x4
+5x3
+6x3
+0x2
+3x2
+2x

+4

Subtract the divisor x2+2x+1, multiplied by
3x4/x2
from the initial polynomial.
3x^2
-1x3
-1x3
-3x2
-2x2
-2x
-1x
Subtract the divisor multiplied by
-x3/x2
from the previous step remainder.
-x
-1x2
-1x2
+3x
-2x
+4
-1
Subtract the divisor multiplied by
-x2/x2
from the previous step remainder.
-1
+5x +5 The remainder degree is 1. It is less than the divisor degree:2. Done.

The division result: 3x2-x-1. The remainder: 5x+5.

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