How do I do synthetic division step by step?

Bring down the first coefficient. Multiply it by r and add that product to the next coefficient; repeat across the row. The last sum is the remainder, and the earlier sums are the quotient coefficients. For x^3 - 6x^2 + 11x - 6 divided by x - 1, the bottom row is 1 -5 6 0, giving x^2 - 5x + 6 and remainder 0. The tool prints the table and every step.

Should I enter r or the divisor, and what about missing powers?

Either form works: 1, x - 1 and (x - 1) mean r = 1; x + 2 means r = -2. You can enter a coefficient list with spaces or commas, or an expanded polynomial. For x^3 - 1 the tool inserts the missing powers as 1 0 0 -1. Dividing by x - 1 gives x^2 + x + 1 and remainder 0.

Can it use the remainder theorem or factor theorem?

Yes, for the r you supply. The remainder is the value of the polynomial at r. A zero on the exact whole-integer path establishes x - r as a factor and returns the remaining quotient. For 2 0 -4 7 divided by x + 2, the result is 2x^2 - 4x + 4 with remainder -1, so the polynomial evaluates to -1 at -2. The calculator does not search for all roots or perform a complete factorization.

Can it divide using decimals, fractions or complex numbers?

Real-number decimals and fractions such as 3/2 are accepted in coefficient lists and in r. Nonintegers switch to floating point, shown above the table, with output at 12 significant digits. For 0.1 0.2 at r = 1, the printed remainder is 0.3, while the underlying sum is 0.30000000000000004. A floating-point zero does not prove an exact factor. Complex numbers, imaginary coefficients and scientific notation are not supported.

Can I enter a nonmonic or trinomial divisor?

This tool accepts the usual synthetic division form x - r. It refuses 2x - 3 and divisors of degree two or more; it does not automatically compute those divisions. For 2x - 3 it explains the algebraic conversion to r = 3/2 and the quotient scaling, but that instruction is not an automatic result for the entered divisor.

How are constant terms and constant polynomials handled?

An ordinary constant term belongs at the end of the coefficient list. A padded constant such as 0 0 5, or a written polynomial such as x^2 - x^2 + 5, is evaluated as the constant 5: division by x - 1 gives quotient 0 and remainder 5. Its degree is 0; the zero quotient has undefined degree. A bare 5 is refused, and a constant divisor is outside the monic linear divisor form.

Does this free calculation program work in a browser, and what are its limits?

Yes. This is browser software with no installation or account needed. It accepts up to 64 coefficients, degree 63, and up to 24 characters per number. Whole-integer arithmetic preserves supported integer inputs exactly: x + 1 at r = 9007199254740993 has remainder 9007199254740994. Copy includes the table and every step. Edited entries, including unfinished text within the input limits, are remembered locally when browser storage is available.

Synthetic Division Calculator

Enter coefficients highest power first or an expanded polynomial, then enter r or a divisor such as x - 1. The calculator shows the synthetic division table, each multiply-add step, the quotient and remainder. It also evaluates the polynomial at the supplied r.

1 -6 11 -6r = 1Exact integers

divide   x^3 - 6x^2 + 11x - 6   by   x - 1every 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 + 6remainder   0        so x - 1 is a factor

Quotient

x^2 - 5x + 6

Remainder

Degree

3 down to 2

x^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.

Enter coefficients highest power first, or an expanded polynomial with one variable. The second box accepts r or a monic linear divisor such as x - 1. The table supports up to 64 coefficients, degree 63. It checks the supplied r; it does not find every root or fully factor a polynomial. Your edited entry is remembered in this browser when storage is available.

1 -6 11 -6 by x - 1r = 1
x^2 - 5x + 6, remainder 0
1 -6 11 -6 by x - 3r = 3
x^2 - 3x + 2, remainder 0
2 0 -4 7 by x + 2r = -2
2x^2 - 4x + 4, remainder -1

Accuracy. Horner arithmetic is exact when every coefficient and r is an integer, using whole-integer arithmetic throughout. Noninteger coefficients or r use floating point, printed to 12 significant digits; a numerical zero remainder does not prove an exact factor. The last number is the polynomial evaluated at r. On the whole-integer path, a zero remainder establishes x - r as a factor.

Common questions

How do I do synthetic division step by step?: Bring down the first coefficient. Multiply it by r and add that product to the next coefficient; repeat across the row. The last sum is the remainder, and the earlier sums are the quotient coefficients. For x^3 - 6x^2 + 11x - 6 divided by x - 1, the bottom row is 1 -5 6 0, giving x^2 - 5x + 6 and remainder 0. The tool prints the table and every step.
Should I enter r or the divisor, and what about missing powers?: Either form works: 1, x - 1 and (x - 1) mean r = 1; x + 2 means r = -2. You can enter a coefficient list with spaces or commas, or an expanded polynomial. For x^3 - 1 the tool inserts the missing powers as 1 0 0 -1. Dividing by x - 1 gives x^2 + x + 1 and remainder 0.
Can it use the remainder theorem or factor theorem?: Yes, for the r you supply. The remainder is the value of the polynomial at r. A zero on the exact whole-integer path establishes x - r as a factor and returns the remaining quotient. For 2 0 -4 7 divided by x + 2, the result is 2x^2 - 4x + 4 with remainder -1, so the polynomial evaluates to -1 at -2. The calculator does not search for all roots or perform a complete factorization.
Can it divide using decimals, fractions or complex numbers?: Real-number decimals and fractions such as 3/2 are accepted in coefficient lists and in r. Nonintegers switch to floating point, shown above the table, with output at 12 significant digits. For 0.1 0.2 at r = 1, the printed remainder is 0.3, while the underlying sum is 0.30000000000000004. A floating-point zero does not prove an exact factor. Complex numbers, imaginary coefficients and scientific notation are not supported.
Can I enter a nonmonic or trinomial divisor?: This tool accepts the usual synthetic division form x - r. It refuses 2x - 3 and divisors of degree two or more; it does not automatically compute those divisions. For 2x - 3 it explains the algebraic conversion to r = 3/2 and the quotient scaling, but that instruction is not an automatic result for the entered divisor.
How are constant terms and constant polynomials handled?: An ordinary constant term belongs at the end of the coefficient list. A padded constant such as 0 0 5, or a written polynomial such as x^2 - x^2 + 5, is evaluated as the constant 5: division by x - 1 gives quotient 0 and remainder 5. Its degree is 0; the zero quotient has undefined degree. A bare 5 is refused, and a constant divisor is outside the monic linear divisor form.
Does this free calculation program work in a browser, and what are its limits?: Yes. This is browser software with no installation or account needed. It accepts up to 64 coefficients, degree 63, and up to 24 characters per number. Whole-integer arithmetic preserves supported integer inputs exactly: x + 1 at r = 9007199254740993 has remainder 9007199254740994. Copy includes the table and every step. Edited entries, including unfinished text within the input limits, are remembered locally when browser storage is available.

Horner arithmetic is exact when every coefficient and r is an integer, using whole-integer arithmetic throughout. Noninteger coefficients or r use floating point, printed to 12 significant digits; a numerical zero remainder does not prove an exact factor.