Synthetic Division

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

1. Arrange the dividend in descending powers and insert zeros for missing coefficients.
2. Set $x-c=0$ and place the resulting $c$ outside the row.
3. Bring down the leading coefficient.
4. Multiply by $c$, place under the next coefficient, and add; repeat.
5. Interpret all but the last number as quotient coefficients and verify remainder using $P(c)$.

Worked example

Representations and interpretation

The synthetic tableau compresses repeated multiply-and-add steps of polynomial long division. A coefficient place-value view explains why missing powers need zero placeholders.

Reasoning about variations

For divisor $2x-3$, basic synthetic division cannot use $3$ blindly; either use $c=3/2$ and account for leading scaling or apply polynomial long division. Naming the method’s condition prevents coefficient errors.

Common mistakes

How to check your work

• Evaluate $P(c)$ and compare with the remainder.
• Multiply divisor by quotient and add remainder.
• Check quotient degree is exactly one less.

Practice

1. Divide $x^3-4x^2+x+6$ by $x-2$.
2. Which synthetic number is used for divisor $x+5$?
3. Why insert a zero coefficient?

Answers and brief solutions

Synthesis and transfer

Testing a proposed zero with synthetic division serves two purposes at once: a zero remainder verifies the factor, and the quotient lowers the degree for further analysis.

Testing $c=3$ in $2x^3-5x^2-4x+3$ through synthetic division yields a zero remainder and quotient $2x^2+x-1$. The quotient factors as $(2x-1)(x+1)$, so the complete roots are $3$, $1/2$, and $-1$. Omitting a zero placeholder for a missing power would shift coefficients and corrupt every later entry. A nonzero final number would equal $P(c)$ by the remainder theorem and reject $c$ as a root while still providing a valid division result. Synthetic division is compact bookkeeping for a specific linear divisor, not a replacement for tracking powers and signs.

Related topics

• Rational Root Theorem
• Roots and Zeros
• Remainder Theorem

Teaching and accessibility note