Polynomial Synthetic Division Calculator: Steps & Examples

This calculator performs synthetic division of a polynomial by a linear divisor of the form (x – c). You enter the coefficients of the dividend polynomial and the value of c, and it returns the quotient polynomial and the remainder, with each step shown clearly. The result is verified against the Remainder Theorem, so you can trust the output for homework, exam prep, or quick checks.

Formula

Synthetic division is a shorthand method for dividing a polynomial P(x) by a linear factor (x – c). The process uses only the coefficients of P(x) and the constant c. If P(x) has degree n, the quotient has degree n–1, and the remainder is a constant.

The algorithm works like this:

• Write down the coefficients of P(x) in descending order of degree.
• Bring down the leading coefficient.
• Multiply it by c and add to the next coefficient.
• Repeat until you reach the last coefficient.
• The last number is the remainder; the others are the coefficients of the quotient.

In symbols, if P(x) = a_nxⁿ + a_n–1x^n–1 + … + a₀, then after synthetic division you get:

P(x) = (x – c)·Q(x) + R

where Q(x) = b_n–1x^n–1 + … + b₀ and R is a constant. The coefficients b_i are computed recursively:

b_n–1 = a_n, b_i = a_i+1 + c·b_i+1 for i = n–2 down to 0, and R = a₀ + c·b₀.

Worked Example

Let’s use the default values: polynomial coefficients 1, –6, 11, –6 (which represent x³ – 6x² + 11x – 6) and c = 1 (so we divide by x – 1).

1. Write the coefficients: 1, –6, 11, –6.
2. Bring down the first coefficient: 1.
3. Multiply 1 by c (1) → 1; add to –6 → –5.
4. Multiply –5 by 1 → –5; add to 11 → 6.
5. Multiply 6 by 1 → 6; add to –6 → 0.

The quotient coefficients are 1, –5, 6, which represent x² – 5x + 6. The remainder is 0. So (x³ – 6x² + 11x – 6) ÷ (x – 1) = x² – 5x + 6 exactly.

Check: (x – 1)(x² – 5x + 6) = x³ – 5x² + 6x – x² + 5x – 6 = x³ – 6x² + 11x – 6. Correct.

When to Use Synthetic Division

Synthetic division is a fast, compact way to divide a polynomial by a linear factor. It is especially useful when you need to:

• Find factors of a polynomial (when the remainder is zero).
• Evaluate a polynomial at a point using the Remainder Theorem.
• Check if a number is a root of a polynomial.
• Factor higher-degree polynomials in algebra or calculus.

However, it only works when the divisor is of the form (x – c). If the divisor is quadratic or higher, you must use long division or another method.

How to Read the Result

The output shows the quotient coefficients in descending order. For example, if the quotient coefficients are 1, –5, 6, that means the quotient polynomial is 1·x² + (–5)x + 6 = x² – 5x + 6. The remainder is a constant; if it is zero, the divisor is a factor of the dividend.

Remember: the degree of the quotient is one less than the degree of the dividend. If the dividend has degree 4, the quotient will have degree 3, and so on.

Common Mistakes and How to Avoid Them

• Missing zero coefficients: If a term is missing (e.g., x³ + 2x – 5 has no x² term), you must include a 0 as a placeholder. Otherwise, the result will be wrong.
• Sign of c: Synthetic division uses (x – c). If you are dividing by (x + 3), then c = –3, not +3. A common error is to use the wrong sign.
• Mixing up quotient and remainder: The last number is the remainder, not part of the quotient. Make sure you separate them correctly.
• Using synthetic division for non-linear divisors: It does not work for (x² – 1) or (2x – 3). For (2x – 3), you must first divide by 2 or use long division.

Limitations and Alternatives

Synthetic division is limited to linear divisors of the form (x – c). If you need to divide by a quadratic or higher-degree polynomial, you have two alternatives:

• Polynomial long division: This is the general method that works for any divisor. It is more tedious but always applicable.
• Factoring and canceling: If the divisor is a factor, you can factor the numerator and cancel, but this requires you to find the factors first.

For evaluating a polynomial at a point, synthetic division is often faster than direct substitution, but the Remainder Theorem gives the same result.

Frequently Asked Questions

What is the difference between synthetic division and long division?

Synthetic division is a shortcut that only works for linear divisors of the form (x – c). It uses only the coefficients and is much faster. Long division works for any divisor, but it involves writing out all the terms and is more time-consuming.

Can synthetic division be used for divisors like (2x – 3)?

Not directly. The divisor must be in the form (x – c). For (2x – 3), you can rewrite it as 2(x – 3/2) and divide by (x – 3/2) using synthetic division, then divide the quotient by 2. Alternatively, use long division.

What does a remainder of zero mean?

A remainder of zero means the divisor (x – c) is a factor of the dividend polynomial. In other words, c is a root of the polynomial. This is a key step in factoring polynomials and solving polynomial equations.

How do I handle missing terms in the polynomial?

You must include a 0 as a placeholder for any missing degree. For example, for x³ – 2x + 1, the coefficients are 1, 0, –2, 1. If you omit the zero, the synthetic division will produce incorrect results.

Is synthetic division the same as the Remainder Theorem?

They are related but not the same. The Remainder Theorem states that the remainder when dividing P(x) by (x – c) equals P(c). Synthetic division gives you both the quotient and the remainder, so you can use it to apply the Remainder Theorem quickly.

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in coeffs | Coefficients (highest degree first) | - | 1,-6,11,-6 | 0
in c | Value of c in (x - c) | - | 1 | 0
out quotient | Quotient coefficients | - | polydiv(coeffs,c) | 2
out remainder | Remainder | - | polyrem(coeffs,c) | 2