Definition: Synthetic division is a shortcut for dividing a polynomial by a linear factor x – c, using only the coefficients and a short row of arithmetic.
When can you use it?
Only when the divisor has the form x – c. You write the coefficients of the polynomial (with a zero for any missing power), put c on the left, then bring down, multiply, and add, column by column.
How does it work?
Worked example: divide 2x3 – 5x2 + 0x + 11 by x – 3. Here c = 3.
- Bring down 2.
- 2 times 3 = 6; -5 + 6 = 1.
- 1 times 3 = 3; 0 + 3 = 3.
- 3 times 3 = 9; 11 + 9 = 20.
The bottom row 2, 1, 3, 20 gives the quotient 2x2 + x + 3 with remainder 20.
What mistakes are common?
Forgetting the placeholder zero for a missing term shifts every column. Using the wrong sign for c is another trap: for x + 4 you must use c = -4. Mathbot checks by plugging c into the original polynomial; the result should equal the remainder.
Example: (2x^3 – 5x^2 + 11) / (x – 3) = 2x^2 + x + 3, remainder 20
Where to learn it
- Dividing Polynomials: Remainder and Factor Theorems – lesson
- Polynomial and Rational Functions: Zeros and Asymptotes – lesson