Synthetic Division Calculator
Type the coefficients highest power first, say what r is, and the table appears the way it is written on paper: the bring-down row, the multiplied row, the rule, and the quotient with the remainder past the bar. Dividing x^3 - 6x^2 + 11x - 6 by x - 1 gives the quotient x^2 - 5x + 6 and remainder 0, and every column is also written out underneath, so 1 times 1 is 1, then -6 plus 1 is -5, which is the part you are usually checking. The coefficient box reads a polynomial written out as well as a list, so x^3 - 1 is understood as 1 0 0 -1: the zeros for the missing powers are inserted for you, and forgetting them is the single commonest way a hand-worked table goes wrong. The r box takes r on its own or the whole divisor, so 1, x - 1 and (x - 1) are the same thing and x + 2 is read as r = -2. When every coefficient and r are whole numbers the arithmetic never leaves whole numbers, so x + 1 divided by x - 9007199254740993 gives the remainder 9007199254740994 exactly, where a calculator working in doubles has already lost the odd number. A decimal or a fraction moves the working to floating point, and the panel says so on the line above the table rather than in a footnote. Up to 64 coefficients, which is degree 63. Nothing is uploaded, no account is asked for, and the only thing kept is your last entry in this browser.
divide x^3 - 6x^2 + 11x - 6 by x - 1
every coefficient and r are whole numbers, so this table is exact
r = 1 | 1 -6 11 -6
| 1 -5 6
+-----------------------
1 -5 6 | 0
quotient x^2 - 5x + 6
remainder 0 so x - 1 is a factorx^3 - 6x^2 + 11x - 6 divided by x - 1 is x^2 - 5x + 6 with remainder 0, so x - 1 is a factor.
Read as 4 coefficients, highest power first: x^3 - 6x^2 + 11x - 6. A power you do not write counts as a zero.
Every row of the table
- Bring downBring down 1.
- Column 21 × 1 = 1, then -6 + 1 = -5.
- Column 3-5 × 1 = -5, then 11 - 5 = 6.
- Column 46 × 1 = 6, then -6 + 6 = 0.
The first box takes the coefficients highest power first, separated by spaces or commas, and it also reads the polynomial written out: x^3 - 1 is understood as 1 0 0 -1, because a power you leave out is the zero the method needs and dropping it is the commonest way a hand-worked table goes wrong. The second box takes r on its own, or the whole divisor, so 1, x - 1 and (x - 1) are the same thing and x + 2 is read as r = -2. Synthetic division is defined for a linear divisor only. A divisor that starts with a number, like 2x - 3, is not one of these: the page says which r to use and which number every quotient coefficient then has to be divided by, since the remainder is the only part that stays put. Up to 64 coefficients, which is degree 63. Whole numbers stay whole numbers of any size; a decimal or a fraction moves the table to floating point and the panel says so above the working. Your entry is kept in this browser so the page opens where you left it, and it goes nowhere else.
Common questions
- How do you do synthetic division step by step?
- Write the coefficients of the dividend in order, highest power first, and put r beside them, where the divisor is x - r. Bring the first coefficient straight down. Multiply what you brought down by r, write the answer under the next coefficient, and add the column. Repeat to the end. The numbers under the rule are the quotient, one degree lower than you started, and the last one is the remainder. For x^3 - 6x^2 + 11x - 6 with r = 1 that is: bring down 1; 1 times 1 is 1, then -6 plus 1 is -5; -5 times 1 is -5, then 11 minus 5 is 6; 6 times 1 is 6, then -6 plus 6 is 0. Quotient x^2 - 5x + 6, remainder 0. This page prints that table and repeats every column in words underneath.
- Is r the divisor, or the number in it?
- r is the number in x - r, not the divisor itself, and the sign is where most of the mistakes happen. Dividing by x - 1 means r = 1. Dividing by x + 2 means r = -2, because x + 2 is x - (-2). You do not have to work that out here: the r box takes either form, so 1, x - 1 and (x - 1) all give r = 1, and typing x + 2 sets r to -2 and says so in the r cell. A fraction is allowed too, so 3/2 and 1.5 are the same entry.
- What if the divisor is not x - r, such as 2x - 3?
- Then plain synthetic division does not apply, and this is exactly where the method quietly changes. Synthetic division is defined for a linear divisor whose x term has a coefficient of 1. For 2x - 3, write it as 2 times (x - 3/2): divide by x - 3/2, which is r = 3/2, and then divide every coefficient of the quotient by 2. The remainder does not change, because it is the value of the polynomial at 3/2 either way. Type 2x - 3 into the r box and the page says precisely that, with the r and the number to divide by filled in for your divisor, and x - 3/2 is then a divisor it reads. A divisor of degree 2 or more, like x^2 - 1, has no synthetic form at all and needs polynomial long division.
- Why do missing terms need a zero?
- Because the table works column by column, and every column stands for one power of x. If a power is missing from the polynomial its coefficient is 0, and leaving the column out shifts every later coefficient into the wrong power. x^3 - 1 is 1 0 0 -1, not 1 -1. You can type either form here: write it out as x^3 - 1 and the zeros are inserted for you, and the coefficients read are shown on the stage so you can see what the page thinks it has. Dividing that by x - 1 gives x^2 + x + 1 with remainder 0.
- What does the remainder tell me?
- The remainder is the polynomial evaluated at r, which is the remainder theorem, so synthetic division is also the fastest way to work out a value of a polynomial by hand. If the remainder is 0 then x - r divides the polynomial exactly and r is a root, which is the factor theorem: 1 -6 11 -6 with r = 1 leaves 0, so x - 1 is a factor and the quotient x^2 - 5x + 6 is what is left to factor. If the remainder is not 0 it is still useful: 2 0 -4 7 divided by x + 2 gives 2x^2 - 4x + 4 with remainder -1, so the polynomial is worth -1 at x = -2 and x + 2 is not a factor.
- Can it divide polynomials with decimals or fractions?
- Yes, and the page is explicit about which arithmetic it used. Whole numbers of any size stay whole numbers from the parse to the printed digit, so nothing rounds. The moment a coefficient or r is not a whole number, the table is worked in IEEE double precision and printed to 12 significant digits, and the line above the table says so before you read a single value. That line matters: 0.1 and 0.2 with r = 1 print a remainder of 0.3, while the double behind it is 0.30000000000000004, so the page flags the row as shortened rather than letting a rounded value pass for an exact one. One consequence is worth knowing: a remainder of 0 on the floating-point path means the doubles cancelled, which is strong evidence that x - r is a factor rather than the proof the whole-number path gives.
- How large a polynomial can it take?
- Up to 64 coefficients, which is a polynomial of degree 63, with each number up to 24 characters long. Past either limit the page says what it counted and what the limit is, rather than cutting the list. Within the whole-number path there is no precision limit at all: x + 1 divided by x - 9007199254740993 comes back with the remainder 9007199254740994, where 9007199254740993 is the first integer a double cannot hold.
Exact Horner arithmetic, kept in whole integers whenever every coefficient and r are integers, so the remainder is the polynomial evaluated at r with no rounding at all. A decimal coefficient switches the working to floating point and the page says so before it prints the table.