The Remainder Theorem says that when a polynomialp(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.
In Spanish
teorema del residuo
Also called
polynomial remainder theorem, little Bezout theorem, remainder theorem for polynomials
The long way (dividing) and the short way (substituting x = 2) give the same remainder, 1. That is the Remainder Theorem.
Formal definition and notation
For every polynomial p(x) and every numbera, there is a unique polynomial q(x) with p(x) = (x − a)q(x) + r, where the remainderr is a constant. Substituting x = a gives p(a) = 0 · q(a) + r = r. A direct consequence is the Factor Theorem: x − a is a factor of p(x) if and only if p(a) = 0.
Symbol
Read it as
Meaning
p(x) = (x − a)q(x) + r
p of x equals x minus a times q of x, plus r
the division of p(x) by x − a, with quotientq(x) and remainder r
r = p(a)
r equals p of a
the Remainder Theorem
x + a = x − (−a)
x plus a equals x minus negative a
dividing by x + a means evaluating at −a
Key facts
Remainder of p(x) ÷ (x − a)= p(a): substitute instead of dividing.
Watch the sign: dividing by x + 4 means substituting x = −4.
Factor Theorem:p(a) = 0 exactly when x − a is a factor, and then a is a root of p(x) = 0.
synthetic division computes p(a) and the quotient at the same time; its last number is the remainder.
The remainder from dividing by a linear divisor is always a constant, because its degree must be less than 1.
Remainder Theorem
Remainder Theorem
p(x) ÷ (x − a) leaves remainder p(a)
p(x)
the polynomial being divided (the dividend)
x − a
a linear divisor with leading coefficient 1
p(a)
the value of the polynomial at x = a
The theorem applies to divisors of the form x − a. For 2x − 3, evaluate at x = 32: the remainder is still p( 32 ).
Remainder Theorem examples, step by step
Example 1: Remainder by substitution
Warm-up
Find the remainder when p(x) = x2 + 3x − 7 is divided by x − 2.
The divisor is x − 2, so a = 2.
p(2) = 4 + 6 − 7 = 3.
Answer: The remainder is 3.
Example 2: A plus sign in the divisor
Core
Find the remainder when p(x) = 2x3 − x2 + 4 is divided by x + 1.
Write x + 1 = x − (−1), so a = −1.
p(−1) = 2(−1)3 − (−1)2 + 4.
= −2 − 1 + 4 = 1.
Answer: The remainder is 1.
Example 3: Find a missing coefficient
Test-style
For what value of k is x − 2 a factor of p(x) = x3 + kx2 − 4x + 12?
By the Factor Theorem, x − 2 is a factor exactly when p(2) = 0.
p(2) = 8 + 4k − 8 + 12 = 4k + 12.
Set 4k + 12 = 0: k = −3.
Check: x3 − 3x2 − 4x + 12 = (x − 2)(x2 − x − 6).
Answer: k = -3
Common mistakes
✗ MistakeRemainder of p(x) ÷ (x + 3) is p(3).
✓ InsteadIt is p(−3).
The theorem uses the number that makes the divisor 0. x + 3 = 0 when x = −3.
✗ Mistakep(4) = 0, so 4 is a factor of p(x).
✓ Insteadp(4) = 0, so x − 4 is a factor and 4 is a root.
The factor is the binomialx − 4; the number 4 is a zero (root) of the polynomial.
✗ MistakeUsing p(a) for the remainder when dividing by x2 − 4.
✓ InsteadFor a quadratic divisor, do long division; the remainder can be linear.
The Remainder Theorem is for linear divisors x − a only.
Not to be confused with
Remainder Theoremvs.Factor Theorem
The Remainder Theorem gives the remainder, p(a), for any a. The Factor Theorem is the special case p(a) = 0: then x − a is a factor.
Synthetic division is a method that finds both the quotient and the remainder. The Remainder Theorem is a fact that finds the remainder alone, by substitution.
Remainder Theoremvs.Remainder
A remainder in arithmetic is what is left after whole-number division: 17 ÷ 5 leaves 2. The Remainder Theorem is about the remainder of dividing polynomials.
Quick check: 3 questions
Question 1 of 3
What is the remainder when x3 − 5x + 1 is divided by x − 3?
Show answer and explanation
Answer: C.p(3) = 27 − 15 + 1 = 13. −11 is p(−3), the wrong sign; 1 is just the constant term; 0 would mean x − 3 is a factor.
Question 2 of 3
Is x + 2 a factor of p(x) = x3 + 8?
Show answer and explanation
Answer: A. For x + 2 evaluate at −2: (−2)3 + 8 = 0, so it is a factor. p(2) = 16 tests x − 2 instead, and p(2) is not 0.
Question 3 of 3
If p(5) = 0, which statement must be true?
Show answer and explanation
Answer: C. By the Factor Theorem, p(5) = 0 means x − 5 is a factor. x + 5 would need p(−5) = 0; the other two statements are unrelated to p(5).
Remainder Theorem: frequently asked questions
What is the Remainder Theorem used for?
It gives a quick way to find the remainder of polynomial division, and it tests possible factors. Instead of dividing x4 − 3x + 2 by x − 1, compute p(1) = 1 − 3 + 2 = 0: the remainder is 0, so x − 1 is a factor. It is also handy for evaluating a polynomial with synthetic division.
Why does the Remainder Theorem work?
Any division can be written as p(x) = (x − a)q(x) + r, where r is a number. If you substitute x = a, the first part becomes (a − a)q(a) = 0, leaving p(a) = r. So the value of the polynomial at a and the remainder are always the same number.
What is the difference between the Remainder Theorem and the Factor Theorem?
The Remainder Theorem says the remainder of p(x) ÷ (x − a) is p(a), whatever its value. The Factor Theorem is the special case where that remainder is 0: then x − a divides p(x) evenly, so it is a factor and a is a root. Each theorem is used to prove and apply the other.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-APR.B.2
Cite this entry
Effortless Math Team. “Remainder Theorem.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/remainder-theorem/
Flashcard
TermRemainder Theoremp(x) ÷ (x − a)
DefinitionThe remainder of dividing a polynomial by x − a is p(a). Example: x2 + 1 divided by x − 3 leaves 10.
PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.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 2RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8Synthetic DivisionSynthetic 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.Algebra · Algebra 2, PrecalculusDivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7BinomialA 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 2Long 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 6FactorA 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 6FactoringFactoring 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 8RootsThe 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 2Zero of a FunctionA zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.Functions · Algebra 1, Algebra 2, PrecalculusTheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, GeometryDegree of a PolynomialThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x3 − 2x5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x2y4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra · Algebra 1, Algebra 2