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Polynomial synthetic division

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Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor (x minus r), using only the polynomial's coefficients instead of full long division. Learn how to set up the grid, bring down and multiply coefficients, and read the quotient and remainder from the result, with a worked example.

What synthetic division is

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x − r). It uses only the coefficients of the polynomial, arranged in a compact grid, instead of writing out the full long-division layout. It gives the same quotient and remainder as long division of polynomials, just faster.

Synthetic division: worked example Dividing x^3 - 4x^2 + x + 6 by x - 2 using synthetic division. Root 2 goes outside; coefficients 1, -4, 1, 6 go across the top. Bring down 1, multiply by 2 to get 2, add to -4 to get -2, multiply by 2 to get -4, add to 1 to get -3, multiply by 2 to get -6, add to 6 to get remainder 0. Quotient: x^2 - 2x - 3, remainder 0. Divide x³ − 4x² + x + 6 by (x − 2) 2 1 −4 1 6 1 2 −2 −4 −3 −6 0 Quotient: x² − 2x − 3 Remainder: 0
Synthetic division of x³ − 4x² + x + 6 by (x − 2): quotient x² − 2x − 3, remainder 0.

Write the divisor's root r (from x − r) to the left, then list the polynomial's coefficients in order across the top — using 0 for any missing power of x. Bring down the first coefficient, multiply it by r, add the result to the next coefficient, and repeat across the row.

The bottom row gives the quotient's coefficients (one degree lower than the original polynomial) followed by the remainder. In the example above, dividing a degree-3 polynomial gives a degree-2 quotient, x² − 2x − 3, with remainder 0 — meaning (x − 2) divides evenly, so x = 2 is a root. This connects directly to the remainder theorem and the factor theorem.

Synthetic division only works when dividing by a linear factor (x − r) with leading coefficient 1. For any other divisor, use full polynomial long division instead.

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