1
Synthetic division is long division with the symbols removed
Every line of the long division by x - c contains the same powers of x in the same columns, so the powers carry no information. Strip them and only the coefficients are left.
Lesson · Polynomials and Factoring · Division and roots
Synthetic Division and the Remainder Theorem
When the divisor is x - c, the long division of “Polynomial Long Division” repeats the same three symbols on every line and can be compressed to one row of numbers. The remainder it leaves is not merely a leftover: it is f(c), which is the fact “The Factor Theorem and Rational Roots” is built on.
divisor must be x - c (degree 1, leading coefficient 1)
c │ a₃ a₂ a₁ a₀ multiply down, add across
f(x) = (x - c)·q(x) + f(c)
remainder = f(c)
The remainder theorem
Core ideas
What this lesson establishes
One shortcut and one theorem. The shortcut saves time; the theorem changes what division is for.
2
It applies only to a divisor x - c
Degree one, leading coefficient one. For x² - x + 2, or for 2x - 1 as written, the compression does not work and “Polynomial Long Division”'s method is the method.
3
The remainder is the function value
Dividing f by x - c leaves a constant remainder, and that constant is f(c). So synthetic division evaluates a polynomial as a side effect, usually with less arithmetic than substituting.
The material
Synthetic division, and what the remainder is
The shorthand, the sign that goes wrong, and the theorem that makes it matter.
Divide any cubic by x - c by long division and the pattern repeats: multiply the divisor by a term, subtract, bring down. Because the divisor is x - c and monic, each quotient coefficient is just the previous line's leading coefficient, and each subtraction becomes an addition once the sign of c is flipped.
Definition · Synthetic division
To divide f(x) by x - c: write c, then the coefficients of f in descending order with zeros for missing powers. Bring down the first coefficient. Multiply it by c, add to the next coefficient, and repeat. The final number is the remainder; the others are the quotient's coefficients, one degree lower.
Example · 2x³ - 5x² + 3x - 7 divided by x - 3
The row is 2, -5, 3, -7 and c = 3. Bring down 2; 2·3 = 6 and -5 + 6 = 1; 1·3 = 3 and 3 + 3 = 6; 6·3 = 18 and -7 + 18 = 11. So the quotient is 2x² + x + 6 and the remainder is 11.
The quotient drops exactly one degree, because dividing a cubic by a linear polynomial gives a quadratic. If your quotient has the same degree as f, you have kept one coefficient too many.
The sign of c
The divisor must be written as x - c, and c is what is SUBTRACTED. For x - 3 we have c = 3. For x + 4 write it as x - (-4), so c = -4. Using 4 there divides by x - 4, which is a different polynomial and gives a wrong answer that looks perfectly well-formed.
Example · x³ + 2x² - 5x + 1 divided by x + 4
Here c = -4 and the row is 1, 2, -5, 1. Bring down 1; 1·(-4) = -4 and 2 + (-4) = -2; (-2)(-4) = 8 and -5 + 8 = 3; 3·(-4) = -12 and 1 + (-12) = -11. Quotient x² - 2x + 3, remainder -11.
The remainder theorem
Theorem · The remainder theorem
For any polynomial f and any number c, the remainder on dividing f(x) by x - c is f(c).
By the division algorithm, f(x) = (x - c)q(x) + r(x) where r = 0 or deg r < 1. In either case r is a constant; call it r.
The identity holds for every value of x, so it holds at x = c. Substituting gives f(c) = (c - c)q(c) + r = 0 · q(c) + r = r.
So the constant remainder is f(c).
□
Check it on the first example: f(3) = 2(27) - 5(9) + 3(3) - 7 = 54 - 45 + 9 - 7 = 11, which is the remainder the synthetic division produced. On the second, g(-4) = -64 + 32 + 20 + 1 = -11, again the remainder.
This is worth pausing on, because it points in two directions. It gives a fast way to evaluate a polynomial — three multiplications and three additions for a cubic, against the powers you would otherwise compute. And it says that a remainder of zero means f(c) = 0, which is where “The Factor Theorem and Rational Roots” begins.
Interactive
Synthetic division
The short table, and the remainder theorem checked
Method
Dividing by x - c synthetically
Read c off the divisor first. That is where the errors are.
1
Check the divisor is x - c
Degree one and monic. x - 3 gives c = 3; x + 4 gives c = -4. For 2x - 1, either factor the 2 out first or use long division.
2
Write every coefficient, zeros included
Descending order, with 0 for each missing power. x³ - 8 is 1, 0, 0, -8, and dropping the zeros is the error that ruins the row.
3
Bring down, multiply, add
Bring the first coefficient down unchanged. Then repeat: multiply the number you just wrote by c, and add it to the next coefficient.
4
Read off the quotient and the remainder
The last number is the remainder, and equals f(c). The rest are the quotient's coefficients, starting one degree below f.
Check yourself
Score 0 / 0Choose the answer you can justify.
2x³ - 5x² + 3x - 7 divided by x - 3
The divisor is x - 3, so c = 3 — the number that makes the divisor zero, not the number printed in it.
3 │ 2 -5 3 -7
│ 6 3 18 multiply down by 3
└──────────────────────────
2 1 6 11
└──── quotient ────┘ remainder
quotient 2x² + x + 6 (degree 2, one below f)
remainder 11
2x³ - 5x² + 3x - 7 = (x - 3)(2x² + x + 6) + 11
check by the remainder theorem:
f(3) = 2(27) - 5(9) + 3(3) - 7 = 54 - 45 + 9 - 7 = 11
The two checks available here are different in kind. Expanding (x - 3)(2x² + x + 6) + 11 confirms the whole division. Evaluating f(3) confirms only the remainder — but it is one line, and a wrong remainder almost always means a wrong row above it.
For a faded row, divide x³ - 4x² + x + 6 by x - 2. The supplied setup is 2 │ 1 -4 1 6. Complete the multiply-add row, name the quotient, and check the last entry with f(2). The bottom row is 1, -2, -3, 0, so q = x² - 2x - 3 and the remainder is 0; using -2 as c computes a different division.
Common mistakes
1
Using the sign printed in the divisorFor x + 4 the value is c = -4. Using +4 performs a valid division by a different polynomial, so nothing looks wrong until the answer is checked — which is the argument for always checking.
2
Leaving out the zero coefficientsx³ - 8 must enter the row as 1, 0, 0, -8. Writing 1, -8 divides some other, lower-degree polynomial and produces a confidently wrong quotient.
3
Reading the remainder as part of the quotientThe last number is separate. For a cubic divided by a linear, the quotient has three coefficients and the fourth number is the remainder; including it makes the quotient a degree too high.
Check your understanding
✓
Finish when you can state the remainder without dividing.
Asked for the remainder when x⁴ - 3x + 1 is divided by x - 2, you should evaluate f(2) = 16 - 6 + 1 = 11 and stop. Reaching for the division at that point means the theorem has not yet replaced the procedure.
Note: Synthetic division is a compression of “Polynomial Long Division”, not a separate theory, and it earns its keep by being cheap enough to run many times. “The Factor Theorem and Rational Roots” tests a list of candidate roots one after another; doing that by long division would be accurate and unusable.