Use our free Synthetic Division Calculator to solve polynomials instantly. Learn the steps, find remainders, and master the Factor Theorem easily.
Synthetic Division Calculator – Step-by-Step Polynomial Solver Do you struggle with dividing polynomials? You are not alone. For many students, long division of polynomials is hard. It takes up a lot of paper. Plus, one…
Do you struggle with dividing polynomials? You are not alone. For many students, long division of polynomials is hard. It takes up a lot of paper. Plus, one small math mistake can ruin your whole answer. If you need to finish homework or study for an exam, you need a better way.
Use our Synthetic Division Calculator. This tool is a fast algebraic shortcut. At My Online Calculators, we want to help you learn. Our tool gives you a full synthetic division solver with clear steps. You will finish your work faster. You will also understand the Factor Theorem and Remainder Theorem better.
Synthetic division is a math hack. It is a quick way to divide polynomials.
Think of it as a shorter version of polynomial long division. The long way forces you to write variables like $x$, $x^2$, and $x^3$ over and over. This is messy. Synthetic division removes the variables. It only uses the coefficients (the numbers in front of the variables). This makes the math clean and fast.
There is one rule. Synthetic division works best when you divide a polynomial, $P(x)$, by a linear binomial like $(x – c)$.
Why do students love this synthetic division calculator? Here are the benefits:
We made our calculator easy to use. It does the hard math for you. Follow these synthetic division steps to get your answer.
Look at your problem. Find the highest exponent. For example, in $3x^4 – 2x^2 + 5$, the highest exponent is 4. Select 4 from the menu. This creates the right number of boxes.
Type the numbers that go in front of your variables. If a term is $x^2$, the coefficient is 1. If it is $-x^3$, enter -1.
This is where most people mess up. If your polynomial skips a degree, you must type a 0. For example, if you have $x^3 – 8$, you skipped $x^2$ and $x$. Your inputs must be:
Find the “Divisor (c)” box. If you divide by $(x – 5)$, enter 5. If you divide by $(x + 4)$, enter -4. Always flip the sign of the number in the binomial.
Click calculate. You will see:
Do you need to show your work on a test? Here are the manual synthetic division steps.
The Problem: Divide $2x^3 – 9x^2 + 13x – 12$ by $x – 3$.
Write 3 on the left. Write the coefficients 2, -9, 13, -12 in a row to the right.
Drop the first number (2) straight down to the bottom row.
Multiply the bottom number (2) by the divisor (3). You get 6. Write 6 under the -9. Add them: $-9 + 6 = -3$. Write -3 on the bottom.
Multiply the new bottom number (-3) by 3. You get -9. Write it under the 13. Add them: $13 + (-9) = 4$. Write 4 on the bottom.
Multiply 4 by 3. You get 12. Write it under the -12. Add them: $-12 + 12 = 0$. The remainder is 0.
Your answer coefficients are 2, -3, 4. This means the answer is $2x^2 – 3x + 4$.
The numbers in our tool prove important math laws.
This theorem says that if you divide a polynomial $P(x)$ by $(x – c)$, the remainder is the same as $P(c)$. This is often faster than plugging the number into a cubic equation calculator or doing the exponent math by hand.
This checks if a binomial is a factor. If the remainder is zero, then $(x – c)$ is a perfect factor. This is a key step in factoring trinomials and larger polynomials.
Which method should you use? Here is a quick comparison.
| Feature | Synthetic Division | Polynomial Long Division |
|---|---|---|
| Input | Only for linear binomials like $(x – 5)$. | Works for any polynomial. |
| Speed | Very Fast. | Slow and messy. |
| Difficulty | Easy. Uses addition. | Hard. Uses subtraction. |
Finding roots (zeros) for high-degree equations is hard. You cannot just use a simple formula. You need a strategy.
You must enter a 0 for any missing term. If you have $x^3 + 1$, you must treat it as $1x^3 + 0x^2 + 0x + 1$.
No. Synthetic division is for linear binomials only. You must use polynomial long division for bigger divisors.
It means the divisor divides perfectly into the polynomial. It is a factor, and the number you tested is a root.
The math is based on $(x – c)$. If you divide by $(x + 3)$, you are really dividing by $(x – (-3))$. So, you use -3.
A synthetic division calculator divides a polynomial by a linear factor such as (x - c) or (x + c). Instead of showing full polynomial long division, it uses only the polynomial’s coefficients and the number c to produce:
It’s the same math you’d do by hand, just faster and with fewer chances to slip on arithmetic.
Synthetic division is meant for linear divisors (degree 1), like:
(x - 3)(x + 2)(2x - 5) (some calculators support this, but you may need an extra step)If the divisor has a higher degree, like (x² + 1), synthetic division isn’t the right tool, and polynomial long division (or another method) is usually required.
(x - c) or (x + c)?You enter the zero of the divisor, meaning the value that makes the divisor equal 0.
Here’s a quick guide:
| Divisor | What you enter |
|---|---|
(x - 5) |
5 |
(x + 5) |
-5 |
(x - 1/2) |
1/2 |
A common mistake is flipping the sign. If your divisor is (x + 2), the number you enter is -2.
Most calculators want the polynomial written normally, or they ask for a list of coefficients. Either way, the big rule is: don’t skip powers.
1, 0, 3, 0, -1Those zeros matter because synthetic division lines everything up by degree.
The output typically gives a row of numbers. The key idea is:
Example idea (no heavy notation): if the calculator returns quotient coefficients 2, -7, 18 and remainder -31, then the quotient is 2x² - 7x + 18, and the remainder is -31.
If your calculator shows only coefficients, make sure you match them to the right powers (they go in order from highest degree down).
A remainder of 0 means the divisor is an exact factor of the polynomial.
It also connects to the Remainder Theorem: when dividing f(x) by (x - c), the remainder equals f(c). So if the remainder is 0, then f(c) = 0, which means c is a root (zero) of the polynomial.
Yes, in a practical way. You can test values of c by dividing by (x - c) and checking the remainder:
c is not a root.Many people use this alongside a graph or a list of possible rational roots to narrow guesses quickly.
Most “wrong” results come from a small input issue:
0 for a missing term)c (entering 2 when the divisor is (x + 2))cIf the remainder doesn’t match what you expect, re-check the polynomial’s order and the divisor sign first. Those two fix most errors fast.