Zero remainder
- Input
- (x^3 - 4x^2 + x + 6) / (x - 2)
- Expected output
- Q = x² − 2x − 3; R = 0
Bottom row: 1, −2, −3, 0. Since the remainder is 0, 2 is a root.
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synthetic division calculator
On the Synthetic division tab, enter P(x) and a degree 1 divisor such as x − 2 or 2x + 1. The table is drawn as in textbooks: root on the left, coefficients on top, quotient and remainder below.
Bottom row: 1, −2, −3, 0. Since the remainder is 0, 2 is a root.
R = P(3) = 54 − 45 + 9 − 7 = 11.
Root −2; coefficients 1, 0, 0, 0, −16.
Root −1; P(−1) = −3 − 2 − 4 = −9.
The table gives 2x² − 2x − 12; divide by 2.
aⁿ − bⁿ is always divisible by a − b.
Divide the leading term of the dividend by the leading term of the divisor, multiply the result by the whole divisor, subtract it from the dividend and repeat with the running remainder until the remainder has a lower degree than the divisor. Example: (2x³ − 3x + 1) ÷ (x² + x − 2). First 2x³ ÷ x² = 2x; 2x·(x² + x − 2) = 2x³ + 2x² − 4x; what is left is −2x² + x + 1. Then −2x² ÷ x² = −2; −2·(x² + x − 2) = −2x² − 2x + 4; what is left is 3x − 3. Quotient 2x − 2, remainder 3x − 3. Missing terms, like the x² in the dividend, go in with a zero coefficient.
Not directly. The method needs a degree 1 divisor, that is, x − a or ax + b. For a divisor like x² + x − 2, use long division, which works for any nonzero divisor.
The last number on the bottom row is P(a). For 2x³ − 5x² + 3x − 7 with a = 3, the bottom row is 2, 1, 6, 11, so P(3) = 11. This avoids computing 3³ and 3² separately, which matters at high degrees and large values.
Because x − a vanishes at x = a. The divisor x − 2 has root +2, and the divisor x + 2 has root −2. The table always uses the value that zeros the divisor, and the tool shows that root in the box on the left.
= 2x³ − 3x + 1
= x² + x − 2
Long division
Division stops when the degree of the remainder is less than the degree of the divisor.
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