Synthetic Division Calculator With Steps
This synthetic division calculator divides a polynomial by a linear factor and shows every step. Enter the dividend, such as x³ − 6x² + 11x − 6, and the divisor, such as x − 2, and it builds the synthetic division table: the coefficients across the top, the products in the middle row and the sums along the bottom. The last number is the remainder and the others are the coefficients of the quotient. It fills in zeros for missing powers, handles divisors like 2x + 1, and checks the answer with the remainder theorem, so you also learn whether x − c is a factor.
Synthetic Division Calculator
Divide a polynomial by x − c with the full synthetic division table, quotient and remainder.
How to use the Synthetic Division Calculator
- Enter the polynomial, like x^3 - 6x^2 + 11x - 6.
- Enter the divisor, like x - 2 (or just 2).
- Read the quotient and remainder.
- Follow the table row by row.
Remainder theorem
When P(x) is divided by x − c, the remainder equals P(c). If P(c) = 0, then x − c is a factor.
Features
- Full synthetic division table
- Zero placeholders for missing terms
- Divisors x − c and ax + b
- Remainder theorem check
- Factor test
Frequently asked questions
How does synthetic division work?
Write the coefficients, bring down the first one, multiply by c, add to the next coefficient, and repeat. The last value is the remainder and the rest are the quotient coefficients.
What is c when dividing by x + 3?
x + 3 = x − (−3), so c = −3. Always use the value of x that makes the divisor zero.
What if a power of x is missing?
Use 0 as its coefficient. For x³ − 1, write 1, 0, 0, −1 before dividing.
Can synthetic division divide by 2x + 1?
Yes. Divide by x + 1/2 using c = −1/2, then divide the quotient coefficients by 2. The remainder stays the same.
What does a remainder of 0 mean?
It means x − c is a factor of the polynomial and c is a root. For example, x³ − 6x² + 11x − 6 divided by x − 2 leaves 0, so 2 is a root.