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Synthetic division is a shorthand method for dividing a polynomial by a linear factor (x − r). It is faster than long division and directly tests whether r is a root of the polynomial.
Formula
Divide polynomial P(x) by (x − c) using a simplified algorithm avoiding long division
- P(x)
- polynomial to divide
- (x − c)
- linear divisor
- c
- value
Guida passo passo
- 1Write coefficients in a row
- 2Drop first coefficient; multiply by root r
- 3Add to next coefficient; repeat
- 4Last value is remainder; others are quotient coefficients
Esempi risolti
Ingresso
x³−6x²+11x−6 ÷ (x−1)
Risultato
Quotient: x²−5x+6, Remainder: 0 (so x=1 is a root)
Ingresso
x³+2x−3 ÷ (x−1)
Risultato
Quotient: x²+x+3, Remainder: 0
Domande frequenti
When is synthetic division useful?
Quick division by linear factors, finding polynomial values via Remainder Theorem, and factoring.
Can synthetic division be used for non-linear divisors?
No, synthetic division only works for divisors of the form (x − c). Use long division otherwise.
What is the Remainder Theorem?
When P(x) is divided by (x − c), remainder = P(c).
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