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Free synthetic division calculator & polynomial synthetic division calculator. Divide polynomials using synthetic division with step-by-step solutions, remainder calculations, and complete algebraic analysis. Perfect for verifying polynomial factors and applying the Factor Theorem.
Last updated: February 2, 2026
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Enter polynomial like x^3 + 2x^2 - 5x + 3
Enter divisor like x - 2 (or just the root like 2)
Quotient:
x^2 + 4x - 3
Remainder:
0
Step-by-Step Solution:
Starting with coefficients: [1, 2, -5, 3]
Dividing by (x - 2), using root = 2
Step 1: 2 + (1 × 2) = 4
Step 2: -5 + (4 × 2) = 3
Step 3: 3 + (3 × 2) = 9
Complete Result:
Polynomial ÷ Divisor = x^2 + 4x - 3 + (0)/(divisor)
Synthetic Division Tips:
Supported polynomials
Quadratic, Cubic, Quartic, and Higher
Works with polynomials of any degree divided by (x - c) or (x + c)
Output format
Quotient + Remainder
Express results as quotient polynomial plus remainder over divisor
Learning feature
Step-by-Step Process
See every calculation with detailed explanations for learning
Remainder Theorem
f(c) = Remainder
Remainder equals the value of polynomial evaluated at the root
Factor verification
Remainder = 0 ⟹ Factor
If remainder is zero, divisor is a factor of the polynomial
Speed advantage
3x Faster than Long Division
Synthetic division is significantly faster for linear divisors
Divide (x³ + 2x² - 5x + 3) by (x - 2) using synthetic division:
Quotient
x² + 4x + 3
Remainder
9
Result: (x³ + 2x² - 5x + 3) ÷ (x - 2) = x² + 4x + 3 + 9/(x - 2)
Our synthetic division calculator uses the efficient synthetic substitution method to divide polynomials by linear binomials. The calculator extracts coefficients, performs the synthetic division algorithm, and presents results with step-by-step explanations.
Step 1: Write coefficients of polynomial in descending order
Step 2: Extract root c from divisor (x - c)
Step 3: Bring down first coefficient
Step 4: Multiply by root, add to next coefficient
Step 5: Repeat until all coefficients processed
Step 6: Last number is remainder, others form quotient
This algorithm is based on Horner's method and significantly reduces the computational complexity compared to polynomial long division when the divisor is linear.
Visual representation of the synthetic division process with coefficients
Synthetic division is grounded in the Division Algorithm for polynomials, which states that for polynomials f(x) and d(x) with d(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that f(x) = d(x)·q(x) + r(x), where the degree of r(x) is less than the degree of d(x).
Need help with other polynomial operations? Check out our polynomial long division calculator and quadratic formula calculator.
Get Custom Calculator for Your PlatformResult:
Quotient: 2x² + 0x + 2 = 2x² + 2
Remainder: 5
(2x³ - 6x² + 2x - 1) ÷ (x - 3) = 2x² + 2 + 5/(x - 3)
Divide x³ - 4x² + x + 6 by (x - 2)
Remainder = 0 ⟹ (x - 2) is a factor!
Find f(4) for f(x) = x³ - 2x² + 5x - 7
Divide by (x - 4), remainder = f(4)
Share it with others who need help with polynomial division
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