Synthetic Division Calculator - Synthetic Division Calculator Polynomial & Polynomial Synthetic Division Calculator
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: December 15, 2024
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Enter polynomial like x^3 + 2x^2 - 5x + 3
Enter divisor like x - 2 (or just the root like 2)
Division Result
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:
- • Synthetic division only works with linear divisors (x - c)
- • If divisor is (x - 2), use root = 2
- • If divisor is (x + 3), use root = -3
- • Remainder of 0 means the divisor is a factor
- • Always write polynomial in descending order of powers
Synthetic Division Calculator Types & Methods
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
Quick Example Result
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)
How Our Synthetic Division Calculator Works
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.
The Synthetic Division Process
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
Mathematical Foundation
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).
- Synthetic division only works for linear divisors (degree 1)
- The Remainder Theorem: f(c) equals the remainder when dividing by (x - c)
- The Factor Theorem: (x - c) is a factor if and only if f(c) = 0
- Missing terms must be represented with coefficient 0
- Synthetic division reduces arithmetic operations by ~60%
- Results can be verified by polynomial multiplication
Sources & References
- College Algebra - Blitzer, Robert F. (8th Edition)Standard reference for polynomial division and synthetic division
- Precalculus: Mathematics for Calculus - Stewart, Redlin, Watson (7th Edition)Comprehensive coverage of synthetic division and factor theorem
- Purplemath - Synthetic Division TutorialEducational resource for learning synthetic division step-by-step
Need help with other polynomial operations? Check out our polynomial long division calculator and quadratic formula calculator.
Get Custom Calculator for Your PlatformSynthetic Division Calculator Examples
Given Information:
- Polynomial: 2x³ - 6x² + 2x - 1
- Coefficients: [2, -6, 2, -1]
- Divisor: (x - 3)
- Root to use: c = 3
Synthetic Division Steps:
- Bring down: 2
- 2 × 3 = 6; -6 + 6 = 0
- 0 × 3 = 0; 2 + 0 = 2
- 2 × 3 = 6; -1 + 6 = 5
Result:
Quotient: 2x² + 0x + 2 = 2x² + 2
Remainder: 5
(2x³ - 6x² + 2x - 1) ÷ (x - 3) = 2x² + 2 + 5/(x - 3)
Factor Theorem Example
Divide x³ - 4x² + x + 6 by (x - 2)
Remainder = 0 ⟹ (x - 2) is a factor!
Remainder Theorem Example
Find f(4) for f(x) = x³ - 2x² + 5x - 7
Divide by (x - 4), remainder = f(4)
Frequently Asked Questions
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