Synthetic Division Calculator
Divide polynomials using synthetic division with step-by-step solutions and explanations
Polynomial Division Setup
Dividend: x^3
Divisor: x - (1)
Division Result
Quotient
Remainder
Division Formula
Step-by-Step Solution
| ×1 | 1 | 0 | 0 | 0 |
| ×1 | 1 | 0 | 0 | 0 |
| 1 |
| ×1 | 1 | 0 | 0 | 0 |
| 1 | ||||
| 1 | 1 |
| ×1 | 1 | 0 | 0 | 0 |
| 1 | 1 | |||
| 1 | 1 | 1 |
| ×1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | ||
| 1 | 1 | 1 | 1 |
Example Problems
Example 1: Monic Divisor
Divide: 3x³ - 8x - 9 by x - 2
Result: 3x² + 6x + 4, R = -1
Example 2: Non-monic Linear
Divide: 4x³ + 2x² - 2x + 1 by 2x + 1
Result: 2x² - 1, R = 2
Synthetic Division Steps
Set up table
Write coefficients and divisor value
Drop leading coefficient
Bring down the first coefficient
Multiply & add
Continue multiply-add pattern
Read results
Last number is remainder
Synthetic Division Tips
Use synthetic division when dividing by linear factors
Much faster than polynomial long division
Great for finding polynomial roots and factors
If remainder is 0, the divisor is a factor
Include zero coefficients for missing terms
Understanding Synthetic Division
What is Synthetic Division?
Synthetic division is a shortcut method for dividing polynomials that uses only the coefficients of the polynomial. It's much faster than polynomial long division and gives the same results.
When to Use Synthetic Division
- •Dividing by linear factors (x - b)
- •Finding polynomial roots and factors
- •Evaluating polynomials at specific values
- •Factoring polynomials completely
Key Concepts
Polynomial Division
P(x) = D(x) × Q(x) + R(x)
Dividend = Divisor × Quotient + Remainder
Factor Theorem
If remainder is 0, then (x - b) is a factor
Useful for finding polynomial roots
Remainder Theorem
P(b) equals the remainder when dividing by (x - b)
Faster than direct substitution