What does a synthetic division calculator do?
A synthetic division calculator runs a shortcut for dividing a polynomial by a linear factor such as x − 3. It drops the variables and works only with numbers, so there is less to write and fewer signs to track.
It gives the same quotient and remainder as polynomial long division. The trade-off is that it only works when the divisor has degree 1.
What number goes in the box?
The box holds the root of the divisor. Set the divisor equal to 0 and solve: for x − 3 the box number is 3, and for x + 2 it is −2.
This is where sign errors start. The box number has the opposite sign of the number in the divisor, because x + 2 is the same as x − (−2). For a divisor like 2x − 1, solve 2x − 1 = 0 to get a box number of 1/2.
Synthetic division steps with an example
Here is (x³ − 2x² − 4) ÷ (x − 3). The dividend has no x term, so its coefficients are 1, −2, 0, −4. The box number is 3.
- Bring down the 1.
- Multiply 1 × 3 = 3 and add to −2 to get 1.
- Multiply 1 × 3 = 3 and add to 0 to get 3.
- Multiply 3 × 3 = 9 and add to −4 to get 5.
- Read the bottom row 1, 1, 3, 5. The last number, 5, is the remainder.
The quotient starts one degree lower than the dividend. A cubic divided by a linear factor gives a quadratic, so 1, 1, 3 means x² + x + 3.
What does the remainder theorem say?
The remainder theorem says that when you divide P(x) by x − c, the remainder equals P(c). So synthetic division doubles as a fast way to evaluate a polynomial.
Check it with the example. P(3) = 27 − 18 − 4 = 5, which is the remainder from above. When the remainder is 0, c is a root and x − c is a factor. That is why the zeros calculator uses synthetic division to test candidates.
How do you use synthetic division with a divisor like 2x − 1?
Use the root 1/2 as the box number, run synthetic division as usual, then divide the quotient row by the leading coefficient 2. The remainder stays as it is.
For (2x³ − 3x² + 4x − 5) ÷ (2x − 1), the row 1/2 | 2, −3, 4, −5 gives the bottom row 2, −2, 3, −7/2. Dividing 2, −2, 3 by 2 gives the quotient x² − x + 3/2, and the remainder is −7/2.
This is the answer the calculator gives, and long division produces the same result. The extra division step is needed because dividing by x − 1/2 and dividing by 2x − 1 differ by a factor of 2.
How do you use synthetic division to factor a polynomial?
Find one root, divide it out with synthetic division, and factor the smaller quotient. Each zero remainder peels off one linear factor.
For x³ − 6x² + 11x − 6, try the box number 1. The row 1 | 1, −6, 11, −6 gives the bottom row 1, −5, 6, 0. The remainder is 0, so x − 1 is a factor and the quotient is x² − 5x + 6. That quadratic factors as (x − 2)(x − 3).
You could start with 2 instead. Dividing by x − 2 gives x² − 4x + 3 with remainder 0, which factors as (x − 1)(x − 3). Any root leads to the same final answer.
When should you use long division instead?
Use long division whenever the divisor is quadratic or higher, such as x² + 1. Synthetic division has only one box number, so it cannot handle a divisor with two roots.
If you enter a divisor like x² + 1 here, the calculator will point you to the polynomial long division calculator. For any linear divisor, synthetic division is the faster choice.
What are the most common synthetic division mistakes?
Using the wrong sign in the box is the classic one. For x + 2, the box number is −2. Leaving out a 0 for a missing power is next: for x³ − 2x² − 4, the row needs 1, −2, 0, −4, not 1, −2, −4.
Some students also subtract instead of add. In synthetic division you always add down the columns, because the sign change is already built into the box number.