Synthetic division is a shortcut for dividing by a linear factor. It is faster than long division and doubles as a way to evaluate P(r) – the remainder theorem says the remainder equals P(r).
How to use the Synthetic Division Calculator
- Degree of P(x)
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀
- Divide by (x − r): r =
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How it works
The leading coefficient is brought down, multiplied by r and added to the next coefficient, and so on. The last number is the remainder; the others are the coefficients of the quotient, one degree lower.
Formula
Bring down a_n; multiply by r and add to the next coefficient; repeat
Remainder = P(r) (remainder theorem)
Worked example
Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): -6; a₁ (x): 11; a₀: -6; Divide by (x − r): r =: 2.
Results — Quotient = x² − 4x + 3; Remainder = 0; Synthetic division tableau = 2 │ 1 −6 11 −6
│ 2 −8 6
──┼ ──── ──── ──── ────
│ 1 −4 3 0; Factor test = (x − 2) is a factor.
Key terms
- Remainder theorem
- P(r) is the remainder when P(x) is divided by x − r.
Frequently asked questions
When can I use synthetic division?
When dividing by a linear binomial x − r. For x + 3 use r = −3.
What does a remainder of 0 mean?
r is a root and (x − r) is a factor.
How do I divide by 2x − 1?
Divide by x − ½ and then divide the quotient by 2.
What is the remainder theorem?
The remainder of P(x) ÷ (x − r) equals P(r).