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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:

PLANETCALC, Polynomial division

Polynomial division

Dividend 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
 
Polynomial division steps

Polynomial division step by step

  • Write down dividend polynomial in a row, including zero terms.
  • Determine first result term by division 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 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 terms sum obtained on the 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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