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Algebra / Polynomial Functions

Synthetic Division

Synthetic division strips polynomial long division down to a single row of numbers. It only works for a linear divisor, but when it applies it is several times faster — and the bottom row hands you the quotient, the remainder, and a function value all at once.

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Once you have done a few polynomial long divisions by a linear factor, you notice that most of the writing is repetition — the ‘s, the subtraction bars, the terms that cancel by design. Synthetic division throws all of it away and keeps only the coefficients. The trade is that it works for one specific shape of divisor, but that shape is exactly the one you meet most often when hunting for zeros.

Synthetic Division — key formula
Key formula

When You Can Use It

Synthetic division applies when the divisor is linear and monic — degree one, with a leading coefficient of :

DivisorUsable directly?Number in the box
yes
yes
yes
not directly — factor out the first, then divide the quotient by
no — use long division—

The box holds the root, not the divisor. You are asking “what value of makes this divisor zero?” For that value is . Writing instead is the error that quietly ruins more synthetic divisions than anything else.

Steps for Synthetic Division

StepWhat you do
1. Set upWrite the root in a box; list the dividend’s coefficients to its right, including zeros for missing powers
2. Bring downCopy the first coefficient straight down to the bottom row
3. MultiplyMultiply that bottom-row number by and write the result under the next coefficient
4. AddAdd the column and write the sum in the bottom row
5. RepeatMultiply-and-add across every remaining column
6. Read offThe last bottom-row number is the remainder; the rest are the quotient’s coefficients, one degree lower than the dividend

The whole method is that one alternation: multiply diagonally, add vertically, all the way across. Written out synthetic division step by step looks like six rules; in practice it is a single rhythm you can run without thinking after a handful of problems.

Synthetic division layout showing the multiply diagonally and add vertically pattern
Multiply down the diagonal, add straight up the column

Synthetic Division Examples

Example 1: An exact division

Divide by . The root is , and the coefficients are .

Step 1 — bring down the leading coefficient:

Step 2 — multiply by the root and add to the next column:

Step 3 — repeat:

Step 4 — repeat once more:

The bottom row is . The final is the remainder; the rest are the quotient’s coefficients:

The dividend was degree and the quotient is degree — dividing by a linear factor always drops the degree by exactly one.

Example 2: A missing term and a non-zero remainder

Divide by . This is the same problem worked by long division in the previous lesson, so you can compare the two side by side.

Step 1 — the divisor is , so the root is . Write the coefficients with a zero for the missing term:

Step 2 — bring down the , then multiply and add:

Step 3 — repeat:

Step 4 — repeat:

Bottom row: .

Identical to the long-division answer, in about a fifth of the writing.

Synthetic division of 2x cubed plus 3x squared minus 5 by x plus 2 giving remainder negative nine
The same division as the long-division lesson, reduced to two rows

Long Division and Synthetic Division Compared

Polynomial and synthetic division answer the same question; they differ in cost and reach.

Long divisionSynthetic division
Divisor allowedany degreeonly
What you writefull polynomials each linecoefficients only
Handles missing powersplaceholder termsplaceholder zeros
Typical length8–12 lines2 rows
Gives you freenoyes — it is the remainder

Use long division when the divisor is quadratic or higher — synthetic polynomial division simply cannot reach those. Use synthetic division whenever you are testing candidate zeros, because you will be doing it dozens of times and the bottom-right number answers “is this a zero?” immediately.

The Bottom-Right Number Is Also P(a)

This is the part that makes synthetic division worth more than its speed. The remainder you get from dividing by is exactly .

In Example 2 the remainder was , so should be . Checking directly:

It matches. That means one pass of synthetic division simultaneously tells you the quotient, the remainder, and the value of the polynomial at — three results for one row of arithmetic. It is why testing a list of candidate roots is fast, and it is the Remainder Theorem in practical form.

Common Mistakes to Avoid

  • Putting the divisor in the box instead of the root. For the box holds . Ask “what makes the divisor zero?” every single time.
  • Omitting the zero for a missing power. has four coefficients — — not three. A missing zero shifts every column and produces a confidently wrong answer.
  • Using it on a non-linear divisor. Dividing by cannot be done synthetically. Long division is the only route.
  • Forgetting the quotient’s degree drops by one. A bottom row of from a cubic means the quotient is , not .
  • Subtracting instead of adding. Long division subtracts; synthetic division adds. The sign flip is already baked into using the root rather than the divisor, and subtracting as well undoes it.

Practice Problems

A set of synthetic division problems, from a routine case to one that asks what the result actually tells you. Work each before opening the answer.

Problem 1. Complete the synthetic division problem: divide by .

Show answer

Step 1 — root is ; coefficients are . Bring down the .

Step 2 — multiply and add:

Step 3 — repeat:

Bottom row: .

Answer: quotient , remainder , so

Problem 2. Divide by .

Show answer

Step 1 — root is ; write the coefficients with both placeholders:

Step 2 — bring down , then multiply and add across:

Bottom row: .

Answer: with remainder

Problem 3. Use synthetic division to find for .

Show answer

Step 1 — divide by , so the root is ; coefficients .

Step 2 — the remainder is , so . Verify directly:

Answer:

Problem 4. A synthetic division by produces the bottom row . What was the original polynomial?

Show answer

Step 1 — the last entry is the remainder ; the rest are the quotient:

Step 2 — rebuild the dividend from :

Answer:

Problem 5. A student divides by and puts in the box, getting a remainder of . What went wrong, and what is the correct remainder?

Show answer

Step 1 — the divisor is , which is zero at , so the box should hold , not .

Step 2 — redo it with the correct root; coefficients :

Step 3 — check against :

Answer: the sign of the root was flipped. The correct remainder is , so is a factor and .

Problem 6. Can you use synthetic division to divide by ? Explain.

Show answer

Step 1 — check the divisor’s form. Synthetic division requires , degree exactly one.

Step 2 — has degree two, so it does not qualify.

You could instead divide twice, since : synthetic division by , then by on the resulting quotient.

Answer: not directly — but two successive synthetic divisions by and achieve the same thing, giving .

The bottom-right number doing double duty as is not a coincidence — it is the Remainder Theorem, which the next lesson makes explicit and then turns into the Factor Theorem. Together they turn synthetic division from a shortcut into a zero-finding tool, which is what makes sketching a polynomial’s end behavior and its middle possible in the same sitting.

Frequently Asked Questions

When can you use synthetic division instead of long division?+

Only when the divisor is linear with leading coefficient — that is, of the form . Dividing by or by does not qualify directly, although can be handled by dividing by and adjusting for the factor of afterwards.

Why do you flip the sign of the number in the box?+

The box holds the *root*, not the divisor. Dividing by means testing , so goes in the box; dividing by means , so goes in. Getting this backwards is the single most common synthetic division error.

How do you do synthetic division with polynomials that have missing terms?+

Write a zero coefficient in place of every missing power before you start. Dividing uses the coefficients — four numbers for a cubic, not three. Skipping the zero shifts every column and gives a confidently wrong answer.

What does the last number in the bottom row mean?+

It is the remainder — and by the Remainder Theorem it also equals , the polynomial evaluated at the number in the box. So a synthetic division that ends in tells you two things at once: the divisor is a factor, and is a zero of the polynomial.

Related lessons

Polynomial Long DivisionFactoring PolynomialsEnd Behavior of Polynomial FunctionsFactoring Higher-Degree Polynomials