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: 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/x2from the initial polynomial. |
3x^2 |
-1x3 -1x3 |
-3x2 -2x2 |
-2x -1x |
Subtract the divisor multiplied by -x3/x2from the previous step remainder. |
-x | ||
-1x2 -1x2 |
+3x -2x |
+4 -1 |
Subtract the divisor multiplied by -x2/x2from 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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