Algebra II

Synthetic Division

Lesson

Synthetic division is a fast shortcut for dividing a polynomial by a linear divisor of the form $x - c$ . It uses just the coefficients — no variables on the page.

It only works when the divisor is $x - c$ with a leading coefficient of 1. For divisors like $x + 3$ , treat the “c” as $-3$ .

Procedure:

Write the dividend coefficients in a row, including 0 for any missing degree.
Write $c$ off to the left.
Bring down the first coefficient unchanged.
Multiply by $c$ , write under the next coefficient, add column.
Repeat until done. The last number is the remainder; the others are the quotient coefficients (one degree lower than the dividend).

Worked example 1

(x^2 + 5x + 6) \div (x + 2)

Use $c = -2$ . Coefficients: 1, 5, 6.

-2 |  1   5   6
   |     -2  -6
   |____________
      1   3   0

Bottom row reads: 1, 3, 0. Quotient is $x + 3$ , remainder is 0.

Worked example 2 — missing term

(x^3 - 7) \div (x - 1)

The dividend is $x^3 + 0x^2 + 0x - 7$ , so coefficients are 1, 0, 0, −7. $c = 1$ .

1  |  1   0   0  -7
   |      1   1   1
   |__________________
      1   1   1  -6

Quotient is $x^2 + x + 1$ , remainder is −6.

How to type your answer

Each problem here divides cleanly (remainder = 0). Type the quotient only, fully simplified, descending order of exponent. Examples: x+3, x^2-2x+5, 2x^2+x-3.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

(x^2 + 5x + 6) \div (x + 2)

Problem 2

(x^2 + 6x + 8) \div (x + 4)

Problem 3

(x^2 - 4x + 3) \div (x - 1)

Problem 4

(x^2 - 25) \div (x - 5)

Practice

Standard problems matching the lesson.

Problem 5

(x^2 + 9x + 14) \div (x + 7)

Problem 6

(x^2 - 7x + 12) \div (x - 3)

Problem 7

(x^2 + x - 12) \div (x - 3)

Problem 8

(x^2 - 36) \div (x + 6)

Problem 9

(x^3 - 8) \div (x - 2)

Problem 10

(x^3 + 1) \div (x + 1)

Problem 11

(x^3 - 6x^2 + 11x - 6) \div (x - 1)

Problem 12

(x^3 - 2x^2 - 5x + 6) \div (x - 3)

Problem 13

(x^3 + 3x^2 - 4x - 12) \div (x + 3)

Problem 14

(x^3 + 4x^2 + x - 6) \div (x + 2)

Problem 15

(2x^3 + 3x^2 - 11x - 6) \div (x - 2)

Problem 16

(x^3 + 27) \div (x + 3)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

(x^4 - 16) \div (x - 2)

Problem 18

(x^4 - 1) \div (x + 1)

Problem 19

(x^3 - 13x + 12) \div (x - 1)

Problem 20

(x^3 + 2x^2 - 9x - 18) \div (x + 2)

Problem 21

(x^4 + x^3 - 8x^2 - 7x + 13) \div (x - 1)

Problem 22

(x^3 - 19x + 30) \div (x - 2)