How this instrument works
Synthetic division is a streamlined shortcut for dividing a polynomial by a linear factor (x−r), skipping the more elaborate bookkeeping of full polynomial long division. For a cubic ax³+bx²+cx+d divided by (x−r), the process runs in three quick steps: the quotient's leading coefficient is simply a; the next coefficient is b plus a·r; the one after that is c plus the previous result times r; and whatever comes out at the very end is the remainder — d plus the prior result times r.
A remainder of exactly 0 carries real meaning: it confirms that r is a genuine root of the original cubic, since dividing evenly with nothing left over means (x−r) is an exact factor. This is exactly how the Factor Theorem gets applied in practice — testing candidate roots one at a time via synthetic division, with a zero remainder confirming a hit.
This page is scoped specifically to a cubic divided by a linear factor, the case where synthetic division's fixed, three-step pattern applies cleanly. Higher-degree polynomials extend the identical multiply-and-add pattern with more steps, and dividing by a non-linear factor needs full polynomial long division instead, since synthetic division's shortcut relies specifically on the divisor being of the simple form (x−r).
- Enter the cubic's four coefficients into the a, b, c, and d fields, from ax³+bx²+cx+d.
- Enter the root of the linear divisor into the Divide by (x − r) field.
- Read the quotient's three coefficients and the remainder: the sheet applies the three-step synthetic division process automatically.
Worked example — x³−6x²+11x−6 divided by (x−1)
Dividing x³−6x²+11x−6 by (x−1): q₂=1, q₁=−6+1×1=−5, q₀=11+(−5)×1=6, and remainder=−6+6×1=0 — the quotient is x²−5x+6, with a remainder of exactly 0, confirming x=1 is a genuine root of the original cubic.
Dividing 2x³+3x²−8x+3 by (x−2) instead gives quotient 2x²+7x+6 with a remainder of 15 — a nonzero remainder means x=2 is NOT a root of this particular cubic, the division simply doesn't come out even.
Questions
What is synthetic division?
A streamlined shortcut for dividing a polynomial by a linear factor (x−r), using a compact series of multiply-and-add steps instead of the more elaborate bookkeeping full polynomial long division requires.
What does a remainder of zero mean?
It confirms that r is a genuine root of the original polynomial — dividing evenly with nothing left over means (x−r) is an exact factor, the Factor Theorem's own defining condition.
Can synthetic division divide by anything other than (x−r)?
Not directly — this shortcut relies specifically on the divisor having that simple linear form. Dividing by a quadratic or higher-degree divisor needs full polynomial long division instead.
How is synthetic division used to find roots?
By testing candidate values of r one at a time — a zero remainder confirms a hit, identifying r as a genuine root, while a nonzero remainder rules that candidate out, a common technique alongside the Rational Root Theorem for narrowing down where a polynomial's roots might be.
Does this extend to polynomials of a higher degree than cubic?
Yes — the identical multiply-and-add pattern extends to any degree, simply with more coefficients and more steps; this page is scoped to the cubic case specifically as the most common starting point.