This synthetic division calculator is provided by Hesapstan to divide a polynomial by a divisor of the form x−r and connect the remainder to the root test.
What does the synthetic division calculator do?
The synthetic division calculator divides a polynomial by a monic linear divisor of the form x−r and shows the full synthetic table.
It returns the quotient, the remainder, and a root-test note that tells whether r is a root of the polynomial.
If the remainder is 0, then P(r)=0 and r is a root. This connects synthetic division to the Remainder Theorem and Factor Theorem.
When synthetic division is valid
Synthetic division is the shortcut version of polynomial division for divisors of the form x−r.
- For x−3, use r=3.
- For x+2, use r=−2.
- The divisor must be monic and linear.
- Divisors such as 2x−3 or x^2+1 require a different method.
x+2 is the same as x−(−2), so the synthetic value is r=−2.
How synthetic division works
Synthetic division uses the polynomial coefficients. The first coefficient is brought down, then each new value is multiplied by r and added to the next coefficient.
- Write the coefficients in descending degree order.
- Bring down the first coefficient.
- Multiply the brought-down value by r.
- Add the result to the next coefficient.
- Repeat until the last coefficient is processed.
The last value is the remainder. The earlier values are the coefficients of the quotient polynomial.
Worked example
Divide P(x)=x^3−6x^2+11x−6 by x−2. Here r=2.
- The coefficient row is 1, −6, 11, −6.
- Bring down 1.
- 1×2=2; add to −6 to get −4.
- −4×2=−8; add to 11 to get 3.
- 3×2=6; add to −6 to get remainder 0.
The quotient is x^2−4x+3 and the zero remainder confirms that x=2 is a root.
How to interpret the remainder
The synthetic-division remainder equals P(r). If it is 0, r is a root; if it is not 0, r is not a root.
A nonzero remainder does not make the division invalid. It simply means x−r is not a factor of the polynomial.
When the remainder is 0, P(x) can be written as (x−r) times the quotient.
Why missing terms matter
A missing degree must be represented by a coefficient of 0. Otherwise, the coefficients shift into the wrong columns.
For example, x^3+2x−5 has no x^2 term, so its coefficient list is 1, 0, 2, −5.
If you omit the zero for a missing degree, the table will divide a different polynomial.
Synthetic division vs polynomial long division
Synthetic division is faster, but narrower. Polynomial long division is the general method for any polynomial divisor of degree at least 1.
- Use synthetic division for x−r.
- Use polynomial long division for x^2+1, 2x−3, or higher-degree divisors.
- Use rational zeros when you need to test many possible r values.
Limitations and common mistakes
The calculator is intentionally limited to monic linear divisors x−r and dividend polynomials of degree at least 1.
- It does not handle non-monic divisors such as ax−r.
- It does not handle quadratic or higher-degree divisors.
- It tests one r value at a time.
- It does not list all possible zeros.
- It requires correct handling of missing middle terms.
Synthetic division here tests the specific r you entered. It is not a complete root-finding algorithm.
Frequently Asked Questions
What divisor can synthetic division use?
Classic synthetic division uses a monic linear divisor of the form x−r.
What is r when the divisor is x+2?
Since x+2 equals x−(−2), the synthetic value is r=−2.
What does a zero remainder mean?
A zero remainder means r is a root and x−r is a factor of the polynomial.
Why do missing terms need zero coefficients?
The zero keeps the coefficient columns aligned with the correct powers of x.
Does this calculator find all roots?
No. It tests one chosen r value. Use the rational zeros calculator to test candidate roots systematically.