What dividing polynomials means
For polynomials with rational coefficients and a nonzero divisor D, there is a unique quotient Q and a unique remainder R satisfying the division identity. The remainder is zero or has lower degree than D. If R = 0, the division is exact and D is a factor of P.
P(x)=D(x)Q(x)+R(x),R=0 or degR<degD OpenStax, College Algebra 2e, §5.4 covers both methods. The identity holds for every x; writing it as P/D = Q + R/D additionally requires D(x) ≠ 0.
Arrange terms without losing coefficients
In 4 + 2x² − x + 3x², the degree 2 terms add to 5x², and the ordered form is 5x² − x + 4. In x³ − 2x² + 4, the degree 1 term is missing, but its place is kept: the coefficients are 1, −2, 0, 4. Omitting that zero in synthetic division would shift the columns and change the polynomial.
Cancellations count too: x² − x² + 0x is the zero polynomial, which has no defined degree. A decimal comma between digits is read as a decimal point; 0,1 represents exactly 1/10.
Why long division works
Dividing the leading term of the partial remainder by the leading term of D gives one term of Q. Multiplying D by that term and subtracting cancels the leading term, so the degree drops with every subtraction and the process ends once the remainder has lower degree than D. The divisor is used as written, even when its leading coefficient is not 1:
x2+1=(2x−4)(2x+1)+5 Dividing the same P by x−2 gives Q=x+2 and the same remainder 5: doubling the divisor halves the quotient and preserves the remainder. For uniqueness, suppose there were two pairs. Then D(Q₁−Q₂)=R₂−R₁. If Q₁−Q₂ were nonzero, the left side would have degree at least that of D, while the right side would be zero or of lower degree: a contradiction.
Synthetic division: the same coefficients with less writing
For D=x−a, synthetic division, also known as Ruffini’s rule, brings down the first coefficient. It then multiplies the last result by a and adds it to the next coefficient. The last sum is R; the preceding sums are the coefficients of Q, which has degree one less than P when P has degree at least 1.
bn=cn,bj=cj+abj+1(j=n−1,…,0) If D=x+1, then a=−1. For x³−2x²+4 divided by x−2, synthetic division yields 1, 0, 0, 4: Q=x² and R=4. Long division handles the same case with a single subtraction, because zero coefficients in Q create no steps; that is why the two methods can take a different number of steps. The RIT polynomial division handout presents both methods.
Synthetic division also works for a divisor ax + b: divide by x − (−b/a), then divide the quotient by a; the remainder stays the same. This calculator handles those divisors with long division.
Remainder and factor theorems
With D=x−a, the remainder is constant. Substituting x=a makes (a−a)Q(a) vanish, leaving P(a)=R. Therefore x−a is a factor of P if and only if P(a)=0.
P(a)=R,(x−a)∣P(x)⟺P(a)=0 For example, x³−6x²+11x−6 divided by x−2 gives Q=x²−4x+3 and R=0. Direct evaluation gives 8−24+22−6=0, confirming the remainder. The calculator also checks every coefficient of DQ+R, rather than only evaluating at a.
That is why synthetic division is used for factoring: try as a the divisors of the constant term (or, when the leading coefficient is not 1, the fractions they form with the divisors of that coefficient), and divide the quotient again with each root found. Each calculation here is one division; to continue, enter the quotient as the new dividend.
Constants, zeros and lower degrees
If P=2x+1 and D=x²+1, the answer is Q=0 and R=2x+1: the dividend already has lower degree than the divisor. If D=2, divide each coefficient by 2 and R=0. Two constants can also be divided: (1/3) divided by (2/5) gives 5/6, with zero remainder.
If P=0 and D≠0, then Q=R=0: every nonzero divisor divides the zero polynomial, and every number is a root of it. If D reduces to zero, as in x−x, division is undefined. In synthetic division, a constant or zero dividend takes just one column: Q=0 and R=P.
Exact fractions and decimal readings
The coefficients are rational numbers: integers, fractions and terminating decimals. Fractions stay reduced, with positive denominators. For example, 0.1x²+0.2x+0.3 divided by x+1 gives Q=(1/10)x+1/10 and R=1/5, without approximating 0.1.
When the quotient or remainder contains fractions, you can view their decimal reading with 4 or 10 significant digits, rounding ties to even. The ≈ sign means the value was rounded: −1/8 = −0.1250 is exact, but 1/3 ≈ 0.3333. Copying the result keeps the complete fractions.
Scope of this tool
Sums of terms in x are accepted up to degree 12. Inner parentheses only group coefficients, so polynomial products must be expanded first.
Finding all factors at once, computing polynomial GCDs, or working with several variables or irrational coefficients are different problems. Paul Dawkins’s Algebra notes at Lamar University provide more division examples. The extended Euclidean algorithm applies successive divisions to integers, and the function analyzer lets you graph and study the associated function.