Synthetic Division
In short: Synthetic division is a compact table that divides a polynomial by x − c using only its coefficients, in place of the full written-out long-division layout.
Algebra 2AlgebraA-APR.D.6
What is Synthetic Division?
Synthetic division trims a polynomial division down to its numbers. For a divisor \(x – c\), line up the coefficients, drop the first one down, then repeat a two-beat rhythm: multiply by \(c\), add into the next column.
The final number of the bottom row is the remainder, and the earlier ones are coefficients of the quotient, one degree lower than the dividend. By the remainder theorem that remainder equals \(p(c)\), so the table doubles as a way to evaluate the polynomial.
How to spot it
- The divisor is a linear factor such as \(x – 4\) or \(x + 2\), with a leading coefficient of 1.
- A problem asks whether \(x – 4\) is a factor, or what \(p(4)\) equals, with little algebra.
- A cubic or quartic must be factored once one root is known, and synthetic division handles the rest quickly.
- The answer is wanted as a quotient plus a remainder, and speed matters.
The method
- List the dividend’s coefficients from the highest power down, put 0 for any missing power, and place \(c\) at the left.
- Drop the first coefficient straight down unchanged.
- Multiply it by \(c\), write the product under the next coefficient, add that column, and repeat across the row.
- Read the bottom row: all numbers but the last build the quotient, and the last number is the remainder.
Synthetic Division: solved example
Problem
Divide \(2x^3 – 7x^2 + 4x + 9\) by \(x – 3\) using the table method.
Solution
Here \(c = 3\) and the coefficients are \(2, -7, 4, 9\). Bring down 2. Multiply: \(2 \times 3 = 6\); add: \(-7 + 6 = -1\). Multiply: \(-1 \times 3 = -3\); add: \(4 – 3 = 1\). Multiply: \(1 \times 3 = 3\); add: \(9 + 3 = 12\).
The quotient is \(2x^2 – x + 1\) and the remainder is 12. Check with the remainder theorem: \(2(27) – 7(9) + 4(3) + 9 = 54 – 63 + 12 + 9 = 12\).
Try it
Question 1
Divide \(x^3 + 2x^2 – 5x – 6\) by \(x + 3\).
Show the answer
Use \(c = -3\) with coefficients \(1, 2, -5, -6\): the bottom row is \(1, -1, -2, 0\). The quotient is \(x^2 – x – 2\) with remainder 0, so \(x + 3\) is a factor.
Question 2
Can the table method divide a polynomial by \(2x – 5\) directly?
Show the answer
No. The divisor must look like \(x – c\) with a leading coefficient of 1; for \(2x – 5\), use long division or first factor out the 2.
Synthetic Division vs. Polynomial Long Division
Both produce the same quotient and remainder, but the layout differs. The table uses only coefficients and one sign flip, and it is limited to divisors of the form \(x – c\). The long version writes every term and handles any divisor. Quick check on the divisor first: linear with leading coefficient 1 means the shortcut is allowed.
Keep going
Lessons
Worksheets and quizzes
- Dividing Polynomials and the Remainder Theorem – Worksheet
- Dividing Polynomials and the Remainder Theorem – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Algebra