# Polynomial division

This calculator divides one polynomial by another polynomial.

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:

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: 3x^{4}+5x^{3}+2x+4 / x^{2}+2x+1.

x^{4} |
x^{3} |
x^{2} |
x | x^{0} |
Description | Result terms |
---|---|---|---|---|---|---|

+3x^{4}+3x ^{4} |
+5x^{3}+6x ^{3} |
+0x^{2}+3x ^{2} |
+2x |
+4 |
Subtract the divisor `x` , multiplied by 3xfrom the initial polynomial. |
3x^2 |

-1x^{3}-1x ^{3} |
-3x^{2}-2x ^{2} |
-2x -1x |
Subtract the divisor multiplied by -xfrom the previous step remainder. |
-x | ||

-1x^{2}-1x ^{2} |
+3x -2x |
+4 -1 |
Subtract the divisor multiplied by -xfrom the previous step remainder. |
-1 | ||

+5x | +5 | The remainder degree is 1. It is less than the divisor degree:2. Done. |

The division result: 3x^{2}-x-1. The remainder: 5x+5.

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