Synthetic Division Calculator

What Is Synthetic Division?

Synthetic division is a streamlined method for dividing a polynomial by a specific kind of linear divisor: (x − r). Instead of writing out every x term like you would in traditional long division, you work almost entirely with coefficients. That makes it fast, tidy, and less error-prone for many algebra tasks—especially when you’re checking possible roots, factoring polynomials, or simplifying expressions in homework, exam practice, and applied problem-solving.

This synthetic division calculator takes the same approach: you enter the coefficients of your polynomial in descending powers, choose a value of r, and the tool returns the quotient polynomial, the remainder, and a full synthetic table you can follow line by line. The goal is not only to compute an answer but to show you the structure behind the method so you can trust the result and learn the pattern.

Why Use Synthetic Division Instead of Polynomial Long Division?

Polynomial long division works for any divisor, including quadratics and higher-degree polynomials. Synthetic division is more specialized: it’s meant for divisors that look like (x − r). That specialization is a strength. It turns a multi-line algebraic process into a repeated “bring down → multiply → add” cycle. If you do this often, synthetic division can save time and reduce mistakes, especially with larger degrees like quartics and quintics.

Another advantage is clarity. A completed synthetic table gives you a clean view of what happened at each step: you see the intermediate products, you see the cumulative sums, and you see the quotient coefficients emerge in the final row. If a result looks unexpected, the table makes it easier to spot where a sign or coefficient might have been entered incorrectly.

How Do You Set Up Synthetic Division?

Setup has two parts: the polynomial and the divisor. The polynomial must be written in descending powers of x, and you must include coefficients for every power—even if some are zero. The divisor must be in the form (x − r). If you have a divisor like (x + 3), rewrite it as (x − (−3)), so r = −3.

Then you arrange the coefficients of the polynomial in a row. Underneath, you build a second row that contains the running results, and between them you insert the products created by multiplying by r. The bottom row becomes your quotient coefficients and remainder.

How Do You Enter Coefficients Correctly?

The most important rule is: enter coefficients in order from the highest power down to the constant term. For example:

• x³ − 3x² + 2x + 5 → 1, -3, 2, 5
• 2x⁴ + 0x³ − 7x + 9 → 2, 0, 0, -7, 9
• −x² + 4 → -1, 0, 4

Notice the zeros. What if you skip them? Your powers won’t align and the synthetic table will represent a different polynomial than you intended. Including zeros is the simplest way to ensure the quotient and remainder match the original expression.

What Happens in Each Step: Bring Down, Multiply, Add

Synthetic division repeats one cycle. First, you “bring down” the leading coefficient to start the bottom row. Next, you multiply that bottom value by r and place it under the next coefficient. Then you add that product to the next coefficient to produce the next bottom value. You keep moving across the row until the last coefficient is processed.

When you’re done, the bottom row has one more entry than the quotient needs. The final entry is the remainder. All earlier bottom entries are the coefficients of the quotient polynomial.

What Is the Remainder Theorem and Why Does P(r) Matter?

A key reason synthetic division is popular is the Remainder Theorem: if you divide a polynomial P(x) by (x − r), the remainder equals P(r). That means you can evaluate a polynomial at r and simultaneously get division information. If the remainder is 0, then P(r) = 0, which tells you r is a root and (x − r) is a factor.

In practical terms, this is a powerful test. What if you suspect that x = 2 is a root? Divide by (x − 2) with synthetic division. If the remainder is 0, you confirmed the root. If it’s not, you immediately know 2 is not a root and you can try another candidate.

What If the Remainder Is Zero?

A zero remainder is the “best case” for factoring. It means your polynomial is divisible by (x − r) with no leftover. The quotient polynomial becomes the remaining factor. For example, if dividing by (x − 1) returns remainder 0 and a quotient of x² − 2x + 5, you can factor:

P(x) = (x − 1)(x² − 2x + 5).

This is especially useful when factoring higher-degree polynomials. Once you peel off one linear factor, you reduce the degree and make the remaining problem much easier.

How Do You Read the Quotient Polynomial From Coefficients?

Suppose your quotient coefficients are 3, -1, 4. If the original polynomial was degree 3, the quotient will be degree 2, and you interpret the coefficients as: 3x² − x + 4.

The degree drops by 1 because dividing by (x − r) removes one power of x. The calculator formats the quotient for you, but it’s still helpful to understand that the coefficient list is a direct map of the polynomial’s structure.

How Does Synthetic Division Help With Factoring Polynomials?

Factoring is often about finding roots. If you can find a value r such that P(r) = 0, then (x − r) is a factor. Synthetic division gives you both the proof (remainder 0) and the reduced polynomial (the quotient). Then you can attempt to factor the quotient further—sometimes by factoring again, by using another synthetic step, or by applying other methods like completing the square or the quadratic formula.

What if you have a polynomial like x³ − 6x² + 11x − 6? Many students try factoring by grouping, but synthetic division quickly tests r values. If r = 1 yields remainder 0, you immediately reduce to a quadratic and finish factoring much faster.

How Do You Choose r Values When You Don’t Know the Root?

If your coefficients are integers and you’re searching for rational roots, a common approach is to test candidates from the Rational Root Theorem. It suggests that possible rational roots are ratios of factors of the constant term and factors of the leading coefficient. You don’t have to memorize the theorem to use synthetic division effectively, but it’s a helpful guide for narrowing down which r values to try first.

If your constant term is small (like ±1, ±2, ±3, ±6, ±12), synthetic division is especially efficient because you can test those values quickly and see whether any produce a zero remainder.

What If Your Polynomial Has Missing Terms?

Missing terms are not a problem as long as you include zeros. For example, if you have: x⁴ + 2x² − 7, the coefficient list must include placeholders for x³ and x: 1, 0, 2, 0, -7.

This alignment prevents “coefficient drift,” where values slide into the wrong power position. The calculator’s preview field is designed to help you sanity-check the polynomial you actually entered before you calculate.

Can You Use Synthetic Division With Decimals or Negatives?

Yes. Synthetic division is arithmetic on coefficients, so it works with negative numbers and decimals. However, decimals can introduce rounding, especially if the division produces repeating decimal patterns. That’s why this tool includes a precision selector. Increase precision when you need more digits for downstream work, or reduce it for a cleaner view while studying.

Why Precision and Number Formatting Matter

Synthetic division results can grow quickly in magnitude, especially with higher degrees or larger r values. Thousands separators can improve readability and reduce copy errors. If you are transferring coefficients into code or another calculator, plain formatting may be better. Choose the option that matches how you plan to reuse the output.

Common Mistakes and How to Avoid Them

Forgetting zeros

This is the most common issue. Always include a zero for every missing power so the coefficient list matches the polynomial’s true structure.

Using the wrong sign for r

Remember: the divisor is (x − r). If the divisor is (x + 4), then r = −4. A sign flip changes every multiplication step and will change the remainder.

Mixing up coefficient order

Coefficients must be entered from highest degree to constant term. If you reverse them, you are dividing a different polynomial than you intended.

How to Interpret the Results in Real Problems

In pure algebra, you typically interpret the remainder as a number. In applied contexts, that number often has meaning: it can represent the “leftover” when a model is simplified, or the value of the polynomial at a particular input. Because P(r) equals the remainder, you can interpret synthetic division as both division and evaluation in one consistent process.

What if you’re building a polynomial model for a dataset and you want to check a specific input value? Using r gives you a quick evaluation of P(r). If you’re analyzing a polynomial’s intercepts, a zero remainder indicates a true root.

When Should You Use Polynomial Long Division Instead?

Synthetic division is best for divisors of the form (x − r). If your divisor is quadratic or higher degree, or if you need to divide by something like (2x − 3) and keep exact coefficient scaling carefully, polynomial long division may be the better tool. Synthetic division can still be adapted in some cases, but the cleanest and most standard use is linear monic divisors.

Examples You Can Try Right Now

• P(x) = x³ − 3x² + 2x + 5, r = 1 (dividing by x − 1)
• P(x) = x⁴ + 2x² − 7, r = 2 (remember zeros in the coefficients)
• P(x) = 2x³ + x² − 8, r = −2 (dividing by x + 2)

Running these examples helps you confirm coefficient entry, sign usage, and remainder interpretation. If you get a remainder of 0, you’ve discovered a factor. If not, the quotient and remainder still give a valid division result: P(x) = (x − r)Q(x) + R.