Polynomial Division & Root Finder
Divide polynomials long-hand or synthetically, find rational roots and differentiate — with the division identity shown holding on every result.
Division
The polynomial itself
What is checked, and what is not claimed
- Polynomial division has an exact identity to be checked against: the dividend must equal the quotient times the divisor plus the remainder, with the remainder of lower degree than the divisor. That either holds or the division is wrong, so it is verified on every result rather than assumed.
- The list of roots holds each one once, however many times it repeats — a list of roots is a list of distinct numbers. Multiplicity is a separate question, and it is answered separately: the factorisation divides by x minus each root again and again for as long as the remainder stays zero, so a double root shows up as a squared bracket rather than being quietly dropped. (x−1)²(x+2) and (x−1)(x+2) have the same roots and are not the same polynomial.
- Roots are found by the rational root theorem — any rational root has its numerator dividing the constant term and its denominator dividing the leading one — and then every candidate is TESTED by evaluation. A near miss never gets through, and irrational or complex roots are simply not claimed.
- Synthetic division is the same arithmetic as long division with the writing stripped away, and only works when dividing by x minus a number. Dividing by a quadratic needs the long form, which is why both are here.
- The product rule is an identity between polynomials rather than an approximation, so it can be checked coefficient by coefficient: the derivative of a product must equal the first times the second derivative plus the second times the first, exactly.
- Coefficients are ordinary floating-point numbers, so dividing by something with awkward coefficients can leave a remainder that should be zero sitting at about a millionth of a millionth instead. Anything smaller than that is treated as zero, which is a choice rather than a fact.
How to use
- Type a polynomial and something to divide it by.
- Read the quotient, the remainder, and the identity that proves them.
- Check the rational roots underneath.
Frequently asked questions
How do I know the division is right?
Because the answer carries its own proof. The dividend must equal the quotient times the divisor plus the remainder, and that identity is worked out and shown on every result — it either holds or the division is wrong, with no middle ground. The same check was swept over 4,000 random divisions.
What is synthetic division and when can I use it?
It is the same arithmetic as long division with the writing stripped away, and it only works when dividing by x minus a number. Dividing by a quadratic needs the long form, so the tool shows the long division always and the synthetic row when it applies.
Why is the last number of the synthetic row the remainder?
That is the remainder theorem: dividing by (x - r) leaves a remainder equal to the polynomial evaluated at r. So if it comes out zero, r is a root and the divisor is a factor — which is why the two are the same question.
How are the roots found?
By the rational root theorem — any rational root has its numerator dividing the constant term and its denominator dividing the leading one — and then every candidate is tested by evaluation. A near miss never gets through, and each root found is confirmed by dividing it out with no remainder.
Why does it say there are no roots when there obviously are?
It only claims RATIONAL roots. x squared minus two has two perfectly real roots and neither is a fraction, so none are listed — and the page says that outright rather than implying the polynomial has none.
Are repeated roots listed twice?
No. Only distinct roots are listed, so a double root appears once. Multiplicity is a separate question and claiming it without saying so would be worse than leaving it out.
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