Math101 · Independent Ontario learning library
Math101Synthetic Division
Synthetic division is a compact algorithm for dividing a polynomial by a linear divisor $x-c$. It operates on coefficients and produces quotient coefficients plus remainder $P(c)$.
Synthetic division tests roots and factors quickly and reduces polynomial degree, enabling exact factorization and equation solving.
Intuition and core definition
Synthetic division is a compact algorithm for dividing a polynomial by a linear divisor $x-c$. It operates on coefficients and produces quotient coefficients plus remainder $P(c)$. It is valid in its basic form only for a monic linear divisor of that form; other divisors require adjustment or long division.
Notation, language, and conditions
Write $c$ beside the synthetic row because $x-c=0$ gives $c$. Include zero coefficients for missing powers. If an $n$th-degree polynomial is divided, the quotient degree is $n-1$, and the final entry is the remainder.
Why this idea matters
Synthetic division efficiently divides by a linear factor $x-c$, producing both quotient coefficients and a remainder equal to the polynomial's value at $c$.
A dependable method
- Arrange the dividend in descending powers and insert zeros for missing coefficients.
- Set $x-c=0$ and place the resulting $c$ outside the row.
- Bring down the leading coefficient.
- Multiply by $c$, place under the next coefficient, and add; repeat.
- Interpret all but the last number as quotient coefficients and verify remainder using $P(c)$.
