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.

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.

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}