Partial Fraction Decomposition Calculator
The partial fraction decomposition calculator decomposes your rational expression with numerator and denominator up to degree 3 into partial fractions (if possible).
Quotientx^2 - 5x + 6
Remainder0
Divides exactly?Yesthe divisor is a factor
Dividendx^3 - 6x^2 + 11x - 6
Divisorx - 1
Remainder theorem check0the dividend evaluated at the divisor’s root
Verification
| divisor × quotient | x^3 - 6x^2 + 11x - 6 |
| plus remainder | x^3 - 6x^2 + 11x - 6 |
| should equal the dividend | x^3 - 6x^2 + 11x - 6 |
The method
dividend = divisor × quotient + remainder
The remainder theorem
Dividing by (x − r) leaves a remainder equal to the polynomial evaluated at r. So a zero remainder means (x − r) is a factor and r is a root — that is the factor theorem, and it is why synthetic division is the standard way to test a candidate root.