Polynomial and Rational Functions

Synthetic Division

Divide x³ - 2x² - 5x + 6 by (x - 3) using synthetic division, produce the quotient polynomial, and verify by multiplication.

The last number is the remainder; the others are quotient coefficients. Example: x³ - 2x² - 5x + 6 ÷ (x - 3), c = 3. Coefficients: 1, -2, -5, 6. Synthetic table: 3 | 1 -2 -5 6 → 1 1 -2 0. Quotient: x² + x - 2, Remainder: 0. Check: (x - 3)(x² + x - 2) = x³ - 2x² - 5x + 6.

Example

Use the synthetic division board to build quotient coefficients and remainder.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

31−2−56□□□□□□□□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & \square & \square & \square \\ \hline & \square & \square & \square & \square \end{array}

Step 2 — Bring down

Bring down the first coefficient: 1.

31−2−56□□□□1□□□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & \square & \square & \square \\ \hline & \hl{1} & \square & \square & \square \end{array}

Step 3 — Multiply by root

Multiply 1 by 3 to place 3 in the next column.

31−2−56□3□□1□□□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & \hl{3} & \square & \square \\ \hline & 1 & \square & \square & \square \end{array}

Step 4 — Add column

Add the column: -2 + 3 = 1.

31−2−56□3□□11□□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & 3 & \square & \square \\ \hline & 1 & \hl{1} & \square & \square \end{array}

Step 5 — Multiply by root

Multiply 1 by 3 to place 3 in the next column.

31−2−56□33□11□□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & 3 & \hl{3} & \square \\ \hline & 1 & 1 & \square & \square \end{array}

Step 6 — Add column

Add the column: -5 + 3 = -2.

31−2−56□33□11-2□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & 3 & 3 & \square \\ \hline & 1 & 1 & \hl{-2} & \square \end{array}

Step 7 — Multiply by root

Multiply -2 by 3 to place -6 in the remainder column.

31−2−56□33-611−2□\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & 3 & 3 & \hl{-6} \\ \hline & 1 & 1 & -2 & \square \end{array}

Step 8 — Add remainder column

Add the remainder column: 6 + -6 = 0.

31−2−56□33−611−20\begin{array}{r|rrrr} 3 & 1 & -2 & -5 & 6 \\ & \square & 3 & 3 & -6 \\ \hline & 1 & 1 & -2 & \hl{0} \end{array}

Step 9 — Result

Read the result: quotient x squared plus x minus 2, remainder 0.

x2+x−2,R=0x^{ 2 }+x- \hl{2} ,\quad R= \hl{0}
synthetic-division Synthetic division efficiently divides a polynomial by a linear factor (x - c). Write the divisor value c and the coefficients of the dividend. Bring down the first coefficient, then multiply and add repeatedly to fill the synthetic table.