Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form x − k. You write only the coefficients, bring down the first one, then repeat two moves: multiply by k and add. The bottom row gives the quotient’s coefficients, and its last number is the remainder, which equals the polynomial’s value at k.
Each column is add, each diagonal is multiply by k. The bottom row, minus its last number, is the quotient; the last number is the remainder.
Formal definition and notation
Let f(x) = anxn + ⋯ + a1x + a0 be a polynomial and x − k the divisor. Synthetic division finds the quotientq(x), of degreen − 1, and the constant remainderr with f(x) = (x − k)q(x) + r. The first quotient coefficient is an; each next one is the next coefficient of f plus k times the previous result, and the final sum is r. By the Remainder Theorem, r = f(k), so r = 0 exactly when x − k is a factor and k is a zero of f. The method needs a divisor of degree 1 with leading coefficient 1.
Symbol
Read it as
Meaning
k
k
the number in the box; for the divisor x − k. For x + 3, k = −3
The remainder is −2, not 0, so by the Remainder Theorem p(−1) = −2.
Answer: No. The remainder is −2, so p(−1) = −2 and x + 1 is not a factor.
Common mistakes
✗ MistakeUsing the coefficients 2, −5, 4 for 2x3 − 5x2 + 4.
✓ InsteadWrite 2, −5, 0, 4, with 0 holding the x place.
Each column stands for one power of x. Skipping a power shifts every later number into the wrong column.
✗ MistakeDividing by x + 3 with 3 in the box.
✓ InsteadUse k = −3, because x + 3 = x − (−3).
The method is built for x − k. The number in the box is the value that makes the divisor zero.
✗ MistakeSubtracting down the columns, as in long division.
✓ InsteadAdd down the columns in synthetic division.
Using k instead of −k already flips the sign, so the subtraction of long division becomes addition.
✗ MistakeReading the bottom row 1, −2, −3 for a cubic as x3 − 2x2 − 3x.
✓ InsteadThe quotient is x2 − 2x − 3.
Dividing by a degree-1 divisor lowers the degree by one, so the first bottom number goes with one power less.
Not to be confused with
Synthetic Divisionvs.Polynomial long division
Long division works for any polynomial divisor, such as x2 + 1, and writes out every term. Synthetic division is a compact shortcut that only works for divisors like x − k.
Synthetic Divisionvs.Remainder theorem
The Remainder Theorem is the fact that dividing f(x) by x − k leaves remainder f(k). Synthetic division is the procedure that finds that remainder quickly.
Factoring rewrites a polynomial as a product. Synthetic division is one tool for it: a remainder of 0 hands you a factor x − k and the other factor q(x).
Quick check: 3 questions
Question 1 of 3
To divide by x + 4 with synthetic division, which number goes in the box?
Show answer and explanation
Answer: A.x + 4 = x − (−4), so k = −4, the value that makes x + 4 zero. 4 forgets to change the sign, and 1 and 14 are not zeros of x + 4.
Question 2 of 3
Synthetic division of a cubic by x − k ends with the bottom row 3, −1, 4, 7. What is the result?
Show answer and explanation
Answer: B. The last number is the remainder, 7, and the others are the quotient’s coefficients starting one degree lower: 3x2 − x + 4. The first choice keeps the old degree, the third treats 7 as a term, and the fourth reads the row backward.
Question 3 of 3
What is the remainder when x3 + 2x2 − 5 is divided by x − 2?
Show answer and explanation
Answer: D. By the Remainder Theorem the remainder is f(2) = 8 + 8 − 5 = 11. −5 is just the constant term, 3 comes from forgetting the 0 for the missing x term, and 27 uses k = 3.
Synthetic Division: frequently asked questions
When can you use synthetic division?
Use it when you divide a polynomial by a linear divisor with leading coefficient 1, such as x − 4 or x + 7. With a small change it also handles ax − b: use k = ba and divide the quotient by a. For divisors of degree 2 or more, such as x2 − 3, use polynomial long division instead.
What does the remainder tell you?
The remainder is the value of the polynomial at k: r = f(k). If it is 0, then x − k is a factor and k is a zero, an x-intercept of the graph. If it is not 0, x − k is not a factor, and you can write the result as q(x) + rx − k.
What do you do with missing terms?
Put a 0 in their place. For x4 − 3x + 2, the powers x3 and x2 are missing, so the row is 1, 0, 0, −3, 2. Every power from the highest down to the constant needs its own column, or the multiply-and-add steps will land in the wrong places.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-APR.B.2, CCSS HSA-APR.D.6
Cite this entry
Effortless Math Team. “Synthetic Division.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/synthetic-division/
Flashcard
TermSynthetic Divisionmultiply by k, add, repeat
DefinitionA shortcut for dividing by x − k using only coefficients. (x2 + 5x + 6) ÷ (x + 2): k = −2, quotient x + 3, remainder 0.
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A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Remainder TheoremThe Remainder Theorem says that when a polynomial p(x) is divided by x − a, the remainder equals p(a). So substituting x = a replaces the division. For p(x) = x2 + 1 divided by x − 3, the remainder is p(3) = 10. A remainder of 0 means x − a is a factor.Algebra · Algebra 2, PrecalculusFactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.Algebra · Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DividendThe dividend is the number being divided in a division problem. In 56 ÷ 8 = 7, the dividend is 56, the divisor is 8 and the quotient is 7. It is the total you start with before splitting it into equal groups. In a fraction such as 568, the dividend is the numerator.Operations · Grade 3, Grade 4, Grade 5PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Long DivisionLong division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.Operations · Grade 4, Grade 5, Grade 6BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusQuadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2