Synthetic Division Calculator: Polynomial Long Division

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Formula at a glance

Synthetic Division Calculator: Division algorithm — P(x) = D(x)·Q(x) + R(x). Example: (2x³ − 3x² + 4x − 5) ÷ (x − 2) → Quotient: 2x² + x + 6 — How the Synthetic Division Calculator works: Division algorithm — P(x) = D(x)·Q(x) + R(x)

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📘 About the Synthetic Division & Polynomial Long Division Calculator

Dividing polynomials works like long division of numbers: P(x) = divisor × quotient + remainder. Synthetic division is a fast shortcut when the divisor is x − c, using only the coefficients. It is used to factor polynomials, find roots and simplify rational expressions.

Enter the dividend and divisor. For a divisor x − c the calculator shows the synthetic table — bring down, multiply by c, add — and checks the remainder theorem P(c) = remainder (a remainder of 0 means x − c is a factor). Any other divisor uses polynomial long division.

🧮 Formula

Division algorithm

P(x) = D(x)·Q(x) + R(x)

Remainder theorem

P(x) ÷ (x − c) leaves remainder P(c)

Factor theorem

P(c) = 0 ⇔ (x − c) is a factor

✏️ Worked examples

(2x³ − 3x² + 4x − 5) ÷ (x − 2)

You enter
Dividend P(x)	2x^3 - 3x^2 + 4x - 5
Divisor	x - 2

You get
Quotient	2x² + x + 6
Remainder	7
Result	(2x³ − 3x² + 4x − 5) ÷ (x − 2) = 2x² + x + 6 + 7/(x − 2)
Remainder theorem	P(2) = 7

Quotient 2x² + x + 6, remainder 7.

(x⁴ − 1) ÷ (x² + 1)

You enter
Dividend P(x)	x^4 - 1
Divisor	x^2 + 1

You get
Quotient	x² − 1
Remainder	0
Result	(x⁴ − 1) ÷ (x² + 1) = x² − 1

Quotient x² − 1, remainder 0.

🧭 How to use this calculator

What you enter

Dividend P(x)
Divisor

What you get

Quotient
Remainder
Result
Remainder theorem

❓ Frequently asked questions

How does synthetic division work?

Write the coefficients, bring down the first, multiply it by c, add to the next coefficient, and repeat. The last number is the remainder; the others are the quotient’s coefficients.

What if a power of x is missing?

Use a 0 for it: x³ − 1 has coefficients 1, 0, 0, −1. The calculator does this for you.

Can synthetic division divide by 2x − 1?

Divide by x − 1/2 and then divide the quotient by 2. The calculator handles any divisor with long division.

÷ Synthetic Division & Polynomial Long Division Calculator