Synthetic Division Calculator

Divide a polynomial by (x − r) using synthetic division. Find the quotient polynomial and remainder faster than long division. Check if r is a root of the polynomial.

∑ Algebra📐 Divide p(x) by (x - r) using synthetic method🔢 Math
Divisor r in (x - r)
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Formula & Reference

VariableSymbolFormulaUnits
Synthetic Division Calculator—Divide p(x) by (x - r) using synthetic methodquotient + remainder

Step-by-Step Examples

Example 1
Divide by (x−2)

Divide x³ − 6x² + 11x − 6 by (x−2). r=2, coeffs: 1,−6,11,−6.

  • Bring down 1
  • 1×2=2; −6+2=−4
  • −4×2=−8; 11−8=3
  • 3×2=6; −6+6=0
  • Remainder=0: x=2 is a root
  • Quotient: x²−4x+3
✓ x²−4x+3, remainder 0
Example 2
Not a Root

Divide x² − 5x + 4 by (x−3).

  • 1,−5,4 with r=3
  • 1; −5+3=−2; 4+(−2×3)=4−6=−2
  • Remainder=−2: x=3 is NOT a root
✓ Quotient: x−2, remainder −2
Example 3
Check Root

Is x=4 a root of x³−4x²+x−4?

  • 1,−4,1,−4 with r=4
  • 1; 0; 1; 0
  • Remainder=0: YES, x=4 is a root
✓ x=4 is a root

Real-World Applications

📚
Algebra
Standard technique taught in precalculus and algebra 2 for polynomial division.
🧪
Engineering
Simplify polynomial fractions in circuit analysis and control theory.
💻
Computer Algebra
CAS systems use polynomial division in simplification algorithms.
📊
Number Theory
Polynomial remainder theorem connects to modular arithmetic.

Common Mistakes to Avoid

⚠️
Forgetting zeros for missing terms

x³ + 2x − 5 has no x² term. Use coefficients: 1, 0, 2, −5. Missing the zero gives wrong quotient.

⚠️
Wrong sign for r

For divisor (x−2), r=2. For (x+3), r=−3. The r is what makes the factor equal to zero.

⚠️
Misreading quotient degree

Dividing a degree-3 polynomial by (x−r) gives degree-2 quotient. Always one degree less than dividend.

Frequently Asked Questions

What is synthetic division? ▾
A shorthand method to divide a polynomial p(x) by (x−r). Works only for divisors of this linear form. Much faster than long division.
How does synthetic division work? ▾
Write coefficients of p(x). Bring down first. Multiply by r and add to next coefficient. Repeat. Final value is the remainder.
What does remainder = 0 tell us? ▾
(x−r) divides p(x) exactly. So r is a root of p(x). This is the Factor Theorem.
Can I use it for divisors other than (x−r)? ▾
No. Synthetic division only works for linear divisors (x−r). For quadratic divisors, use polynomial long division.
What is the relationship to the Remainder Theorem? ▾
The remainder when dividing p(x) by (x−r) equals p(r). Synthetic division computes this remainder quickly without full evaluation.

Related Math Calculators

∑ Polynomial∑ Quadratic∑ Completing the Square∑ Linear Equation∑ Function Composition

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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