Remainder Theorem, Factor Theorem and Bézout’s Theorem

The remainder theorem says that dividing a polynomial p(x) by x − r leaves the remainder p(r), and the factor theorem says that x − r is a factor of p(x) exactly when p(r) = 0. In brief: for p(x) = x³ − 2x² − 5x + 6, p(1) = 0, so x − 1 is a factor. Bézout’s theorem in this setting is the same statement, named after the French mathematician Étienne Bézout.

These theorems connect three ideas: the value of a polynomial at a point, the remainder of a division and the factors of the polynomial. They allow you to test a candidate zero in seconds without doing a division, and they are the basis of the search for roots. This guide states the theorems, proves them, shows how to use them and gives a table of examples.

What does the remainder theorem say?

If a polynomial p(x) is divided by x − r, the remainder is p(r). You can find the remainder by simple substitution, without division. For p(x) = 2x³ + 3x² − 5x + 4 divided by x + 2, take r = −2: p(−2) = 2(−8) + 3(4) + 10 + 4 = −16 + 12 + 10 + 4 = 10.

For a divisor of the form ax − b the remainder is p(b/a). For example, dividing by 2x − 1 gives the remainder p(1/2).

What does the factor theorem say?

The expression x − r is a factor of p(x) if and only if p(r) = 0. That is the special case of the remainder theorem when the remainder is zero. It also says that every zero of a polynomial gives a linear factor, and every linear factor gives a zero.

Example: p(x) = x³ − 2x² − 5x + 6. Try r = 1: 1 − 2 − 5 + 6 = 0, so x − 1 is a factor. Dividing gives x² − x − 6 = (x − 3)(x + 2). The full factorization is (x − 1)(x − 3)(x + 2).

What is Bézout’s theorem for polynomials?

In the context of polynomial division, the name Bézout’s theorem is used in many textbooks for the remainder theorem: the remainder of the division of p(x) by x − r is the number p(r). In other contexts, Bézout’s theorem refers to a different, more general statement about the number of intersection points of two algebraic curves, which says that two plane curves of degrees m and n without a common component meet in at most mn points. In number theory, Bézout’s identity says that gcd(a, b) = ax + by for some integers x and y. Check which meaning your course intends. Here the polynomial version is the one used.

Why do the theorems hold?

Dividing p(x) by x − r gives p(x) = (x − r)q(x) + R, where the remainder R is a constant, since its degree is less than the degree of x − r, which is 1. Substitute x = r: p(r) = (r − r)q(r) + R = 0 + R = R. So the remainder equals p(r). The factor theorem follows: R = 0 exactly when p(r) = 0.

How do you use the theorems?

Example 1: finding a remainder quickly

What is the remainder of x¹⁰⁰ + 3x + 2 divided by x − 1? It is p(1) = 1 + 3 + 2 = 6. A direct division would take very long.

Example 2: finding an unknown coefficient

For which k is x − 2 a factor of x³ − kx² + 4x − 8? Require p(2) = 0: 8 − 4k + 8 − 8 = 0, so 8 − 4k = 0 and k = 2. Check: x³ − 2x² + 4x − 8 = x²(x − 2) + 4(x − 2) = (x − 2)(x² + 4) ✓.

Example 3: two remainders

A polynomial p leaves remainder 5 when divided by x − 1 and remainder 3 when divided by x + 1. Find the remainder when dividing by x² − 1. The remainder has the form ax + b. From p(1) = 5: a + b = 5. From p(−1) = 3: −a + b = 3. Adding gives b = 4 and a = 1. The remainder is x + 4.

Example 4: testing candidates

To factor x³ + 2x² − 5x − 6, test divisors of the constant term 6. p(1) = −8, p(−1) = 0 ✓. So x + 1 is a factor. The quotient is x² + x − 6 = (x + 3)(x − 2).

Remainder test table

Where do the theorems come from?

The remainder theorem is usually attributed to Étienne Bézout, an eighteenth-century French mathematician who wrote a famous series of textbooks, and he is also known for the theorem on the intersection of curves. The factor theorem is essentially the algebraic form of the idea, used by Descartes in the seventeenth century, that each root gives a linear factor. Some sources give credit to Ruffini for the related division scheme.

Where are they used today?

Computer algebra systems use the theorems for factoring and for testing candidate roots. Polynomial remainders are used in coding theory, for example in cyclic redundancy checks, where a message is treated as a polynomial and its remainder is a checksum. In school algebra, the theorems are the fastest way to check a zero.

Which mistakes are common?

Using r with the wrong sign

For divisor x + 3 the value to test is p(−3), not p(3).

Thinking p(r) is the quotient

p(r) is the remainder, which is a number. The quotient is a polynomial found by division.

Applying the theorem to non-linear divisors

For divisors of higher degree, the remainder is a polynomial, not a single value. Use long division or a system of equations as in Example 3.

Mixing up the different Bézout results

The polynomial remainder theorem, Bézout’s identity for integers and Bézout’s theorem on curves are three different statements.