01 / 20

Synthetic
Division

A fast, streamlined method for dividing polynomials by linear factors of the form \((x - c)\).

Algebra 2

What Is Polynomial Division?

Just as we divide numbers to find quotients and remainders, we can divide polynomials. Given a polynomial \(f(x)\) and a divisor \(d(x)\), we find a quotient \(q(x)\) and remainder \(r\) such that:

When the divisor is a linear factor \((x - c)\), we can use a shortcut called synthetic division instead of long division.

Why Use Synthetic Division?

★ Speed
Much faster than polynomial long division: only simple multiplication and addition.
★ Simplicity
Work only with coefficients: no variables to track, fewer chances for error.
★ Versatility
Evaluate \(f(c)\), test roots via the Remainder and Factor Theorems, and reduce degree.

Key Vocabulary

1
Dividend: the polynomial being divided, e.g. \(2x^3 + 3x^2 - 5x + 7\).
2
Divisor: the linear factor \((x - c)\). In synthetic division, we use the value \(c\).
3
Quotient: the result polynomial, one degree lower than the dividend.
4
Remainder: the constant left over. If it equals \(0\), then \((x - c)\) is a factor.

The Remainder Theorem

This means synthetic division does double duty: it finds the quotient and simultaneously evaluates \(f(c)\). If \(f(c) = 0\), then \((x - c)\) divides evenly.

Setting Up the Synthetic Division

To divide \(f(x)\) by \((x - c)\), arrange the coefficients in a row and place \(c\) to the left:

\(c\)
\(a_n\)
\(a_{n-1}\)
 ...
\(a_1\)
\(a_0\)
 divisor value coefficients of f(x), highest to lowest degree

Synthetic vs Long Division

Synthetic Division Polynomial Long Division
Divisor type Linear \((x - c)\) only Any polynomial
Operations Multiply & add Multiply, subtract, bring down
Writes variables? No: coefficients only Yes: full polynomial terms
Speed Fast Slower
Error risk Lower Higher (sign errors)

How to Apply Synthetic Division

1
Identify \(c\): If dividing by \((x - c)\), use \(c\). If dividing by \((x + 3)\), use \(c = -3\).
2
List coefficients: Write all coefficients in descending degree order. Insert \(0\) for missing terms.
3
Bring down the leading coefficient to the bottom row.
4
Multiply & add: Multiply the bottom value by \(c\), write it under the next coefficient, then add. Repeat.
5
Read the result: Bottom row gives quotient coefficients (one degree lower) and the last number is the remainder.

Example 1: \((x^3 + 2x^2 - 5x + 3) \div (x - 1)\)

Divide by \((x - 1)\), so \(c = 1\). Coefficients: \(1, 2, -5, 3\).

1 1 2 -5 3 1 3 -2 1 3 -2 1
\(x^2\)
\(x\)
 const remainder

Example 1: Step-by-Step Trace

1
Bring down the leading \(1\).
2
Multiply: \(1\) \(\times\) \(1\) = \(1\). Add: \(2 + 1 =\) \(3\).
3
Multiply: \(1\) \(\times\) \(3\) = \(3\). Add: \(-5 + 3 =\) \(-2\).
4
Multiply: \(1\) \(\times\) \(-2\) = \(-2\). Add: \(3 + (-2) =\) \(1\) (remainder).

Example 2: \((3x^3 - 4x + 2) \div (x + 2)\)

Dividing by \((x + 2) = (x - (-2))\), so \(c = -2\). The \(x^2\) term is missing, so insert a \(0\): coefficients are \(3, 0, -4, 2\).

-2 3 0 -4 2 -6 12 -16 3 -6 8 -14
\(x^2\)
\(x\)
 const remainder

Example 3: Exact Division (Zero Remainder)

Divide \(2x^3 - 3x^2 - 5x + 6\) by \((x - 1)\). Use \(c = 1\).

1 2 -3 -5 6 2 -1 -6 2 -1 -6 0
\(x^2\)
\(x\)
 const remainder

Reading the Bottom Row

If the dividend has degree \(n\), the quotient has degree \(n - 1\). The bottom row gives the quotient coefficients left to right, starting from \(x^{n-1}\).

Quotient Coefficients
All numbers in the bottom row except the last are the coefficients of the quotient polynomial.
Remainder
The last number in the bottom row is the remainder. By the Remainder Theorem, this equals \(f(c)\).

Finding Roots with Synthetic Division

Synthetic division is a powerful tool for factoring polynomials completely. Once you find a root \(c\) (remainder = 0), the quotient is a reduced polynomial you can factor further.

1
Test a candidate root \(c\) using synthetic division.
2
If remainder = \(0\), then \(c\) is a root and \((x - c)\) is a factor.
3
Factor the quotient further (by synthetic division again, quadratic formula, or inspection).

Common Mistakes

✕ Wrong sign for \(c\)
Dividing by \((x + 3)\) but using \(c = 3\) instead of \(c = -3\). Always write \((x + 3) = (x - (-3))\).
✕ Forgetting zero coefficients
Missing \(x^2\) term in \(x^3 + 5\) means the coefficients must be \(1, 0, 0, 5\), not \(1, 5\).
✕ Subtracting instead of adding
Synthetic division uses addition, not subtraction. Long division subtracts: don't mix them up.

Summary Cheat Sheet

Concept Key Point
When to use Divisor must be linear: \((x - c)\)
Value of \(c\) \((x+3) \Rightarrow c=-3\)
Missing terms Insert \(0\) as placeholder
Operation Multiply by \(c\), then add
Quotient degree One less than the dividend
Remainder = 0 \((x - c)\) is a factor
Remainder Theorem Remainder = \(f(c)\)

Question 1 of 4

When dividing by \((x + 4)\) using synthetic division, what value of \(c\) do you use?

Question 2 of 4

What are the coefficients you set up for \(x^4 - 3x^2 + 7\)?

Question 3 of 4

Use synthetic division to divide \(x^3 + 6x^2 + 11x + 6\) by \((x + 1)\). What is the quotient?

Question 4 of 4

After synthetic division of \(2x^3 + x^2 - 8x - 4\) by \((x - 2)\), the bottom row reads: \(2, 5, 2, 0\). What does the \(0\) tell us?