Synthetic Division Calculator: Divide a Polynomial by (x - c) with Quotient and Remainder
Quick division of a polynomial by x - c.
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How this calculator works
Type the coefficients from the highest power down, separated by spaces.
Enter the value of c for the divisor x - c.
The calculator gives the quotient, the remainder and whether c is a root.
Formulas used
- b_0 = a_0,\ b_i = a_i + c\,b_{i-1}
- P(x) = (x - c)Q(x) + R
- R = P(c)
Worked example
With the default numbers in the calculator above:
- Bottom row of the synthetic division: 1, -2, 3, 6
- Remainder (the value of the polynomial at c): 6
- Quotient coefficients: 1, -2, 3
- Quotient as a polynomial: x^2 - 2x + 3
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: Divide by (x - c): the value of c 1.
- Bottom row of the synthetic division: 1, -5, 6, 0
- Remainder (the value of the polynomial at c): 0
- Quotient coefficients: 1, -5, 6
- Quotient as a polynomial: x^2 - 5x + 6
Common questions
How does synthetic division work?
You bring down the first coefficient, multiply by c and add to the next coefficient, and repeat. The last number is the remainder and the rest are the quotient.
What is the remainder theorem?
The remainder when a polynomial is divided by x - c equals the value of the polynomial at x = c.
How do I divide by x + 2?
Write x + 2 as x - (-2) and use c = -2.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any algebra text (synthetic division, the remainder theorem).
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