Quick start
- Type the coefficients of P(x) from the highest power down. For 2x³ − 3x² + 5, type 2, -3, 0, 5 (the 0 stands for the missing x term).
- Enter a. For division by x − 2, a = 2. For x + 2, a = −2.
- Read P(a), the synthetic division row and the quotient.
The theorem in one line
When a polynomial P(x) is divided by x − a, the remainder is P(a). So instead of doing full long division, you can just substitute.
Example: 2x³ − 3x² + 5 divided by x − 2
P(2) = 2·8 − 3·4 + 0 + 5 = 16 − 12 + 5 = 9, so the remainder is 9. Synthetic division confirms it: bring down 2; 2·2 = 4, −3 + 4 = 1; 1·2 = 2, 0 + 2 = 2; 2·2 = 4, 5 + 4 = 9. The quotient is 2x² + x + 2.
The factor theorem
If P(a) = 0, the remainder is zero and x − a is a factor of P(x). This is a fast way to test possible factors before you divide.
Common questions
How do I enter x + 3?
x + 3 is x − (−3), so enter a = −3.
What if I am dividing by 2x − 1?
Use a = 1/2 (0.5). The remainder is the same, but the quotient must be divided by 2.
Why put 0 for missing terms?
Each coefficient holds a place for one power of x. Skipping one shifts every term to the wrong power.