Polynomial Division

The division algorithm

Let $P ( x )$ and $D ( x )$ be polynomials in $\mathbb{R} [ x ]$ with $D ( x ) \neq 0$. The division algorithm asserts the existence of unique polynomials $Q ( x )$ and $R ( x )$ in $\mathbb{R} [ x ]$ such that:

• The polynomial $P ( x )$ is referred to as the dividend.
• $D ( x )$ as the divisor.
• $Q ( x )$ as the quotient.
• $R ( x )$ as the remainder.

If $R ( x ) = 0$, the division is exact and $D ( x )$ divides $P ( x )$ in $\mathbb{R} [ x ]$. This result is directly analogous to the Euclidean division of integers and holds in any polynomial ring $F [ x ]$, where $F$ is a field. The existence of such a representation can be established by induction on $deg ⁡ P$, while uniqueness follows from a degree argument. Suppose that two representations exist:

Subtracting yields $( Q_{1} ( x ) - Q_{2} ( x ) ) \cdot D ( x ) = R_{2} ( x ) - R_{1} ( x )$. If $Q_{1} \neq Q_{2}$, the left-hand side has degree at least $deg ⁡ D$, whereas the right-hand side satisfies $deg ⁡ ( R_{2} - R_{1} ) < deg ⁡ D$, a contradiction. Therefore $Q_{1} = Q_{2}$, and consequently $R_{1} = R_{2}$.

A ring is defined as an algebraic structure with two operations, addition and multiplication, that satisfy associativity, distributivity, and the existence of additive inverses. A field is a ring in which every nonzero element possesses a multiplicative inverse. Common examples of fields are $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$.

Let $P ( x )$ and $D ( x )$ be nonzero polynomials such that $deg ⁡ P \geq deg ⁡ D$. The degrees of the quotient and remainder are as follows:

\(deg ⁡ Q ( x ) = deg ⁡ P ( x ) - deg ⁡ D ( x )\) \(deg ⁡ R ( x ) < deg ⁡ D ( x )\)

If $deg ⁡ P < deg ⁡ D$, then the quotient is the zero polynomial and the remainder is $P ( x )$.

Polynomial long division

The long division algorithm involves repeatedly dividing the leading term of the current remainder by the leading term of $D ( x )$, subtracting the resulting product, and continuing this process until the degree of the remainder is less than that of $D ( x )$. The procedure can be summarized in the following steps:

• First, divide the leading term of the current dividend by the leading term of $D ( x )$ to determine the next term of $Q ( x )$.
• Next, multiply this term by $D ( x )$ and subtract the result from the current dividend.
• Repeat these steps until the degree of the remaining expression is strictly less than $deg ⁡ D ( x )$.

When the divisor is a linear polynomial of the form $x - c$, the procedure can be carried out more efficiently using the synthetic division method, which reduces the computation to operations on coefficients alone.

Example 1

We apply the method outlined above to compute the quotient of $P ( x )$ and $D ( x )$ where:

\(P ( x ) = x^{3} + 2 x^{2} - x - 2\) \(D ( x ) = x - 1\)

The division table is constructed with terms arranged in descending order of degree:

Dividing the leading term $x^{3}$ by $x$ yields $x^{2}$. Next, multiply $x^{2}$ by $D ( x ) = x - 1$ and subtract the result:

Dividing $3 x^{2}$ by $x$ yields $3 x$. Multiply and subtract as before:

Dividing $2 x$ by $x$ yields $2$. Multiply and subtract to complete the process:

Since the remainder is zero, the division is exact. The result is:

and therefore:

Example 2

The following example demonstrates a case where the division is not exact; specifically, the remainder $R ( x )$ is a nonzero polynomial whose degree is strictly less than $deg ⁡ D ( x )$.

The method outlined above is applied to compute the quotient of the following polynomials:

\(P ( x ) = x^{3} + x^{2} + x + 2\) \(D ( x ) = x^{2} + 1\)

The division table is constructed with terms arranged in descending order of degree:

Dividing $x^{3}$ by $x^{2}$ yields $x$. Multiplying $x$ by $D ( x )$ and subtracting produces the next row:

Dividing $x^{2}$ by $x^{2}$ yields $1$. Multiplying and subtracting results in the following:

The degree of the remainder $1$ is $0$, which is strictly less than $deg ⁡ D ( x ) = 2$. Therefore, the algorithm terminates. The quotient and remainder are:

The division can be written as:

The remainder theorem and the factor theorem

The division algorithm leads to a result that connects polynomial division with the evaluation of a polynomial at a specific point. Instead of computing $P ( c )$ directly, the same value can be obtained as a consequence of dividing $P ( x )$ by the linear polynomial $x - c$. Let $P ( x ) \in \mathbb{R} [ x ]$ and $c \in \mathbb{R}$. When $P ( x )$ is divided by the linear polynomial $x - c$, the remainder equals $P ( c )$.

Applying the division algorithm with divisor $D ( x ) = x - c$, since $deg ⁡ D = 1$, the remainder $R ( x )$ must satisfy $deg ⁡ R < 1$, implying that $R ( x )$ is a constant, denoted $r$. The division algorithm then yields:

Substituting $x = c$ into both sides:

Thus $r = P ( c )$, which establishes the result.

The remainder theorem offers a direct method for evaluating a polynomial at a specific point without performing the complete division. The value $P ( c )$ is given by the remainder when dividing by $x - c$.

The following result, known as the factor theorem, is a direct consequence of the remainder theorem. Let $P ( x ) \in \mathbb{R} [ x ]$ and $c \in \mathbb{R}$. The polynomial $x - c$ divides $P ( x )$ in $\mathbb{R} [ x ]$ if and only if $P ( c ) = 0$.

This result can be proved directly using the remainder theorem. Dividing $P ( x )$ by $x - c$ gives $P ( x ) = Q ( x ) ( x - c ) + r$, where $r = P ( c )$. Therefore, $x - c$ divides $P ( x )$ if and only if $r = 0$, which is equivalent to $P ( c ) = 0$.

The factor theorem establishes a correspondence between the roots of a polynomial and its linear factors: $c$ is a root of $P ( x )$ if and only if $x - c$ is a factor of $P ( x )$ in $\mathbb{R} [ x ]$.

This principle underlies the factorisation of polynomials over a field and will be explored further in the section on polynomial factorisation.

Example 3

As an application of the remainder theorem, consider the polynomial:

and the value $c = 2$. According to the theorem, dividing $P ( x )$ by $x - 2$ yields a remainder equal to $P ( 2 )$. Computing $P ( 2 )$ directly by substitution we obtain:

The remainder theorem predicts that the remainder of dividing $P ( x )$ by $x - 2$ is $1$. This result can be verified using the long division method:

Dividing the leading term $2 x^{3}$ by $x$ yields $2 x^{2}$, which serves as the initial term of the quotient. Multiplying $2 x^{2}$ by $D ( x ) = x - 2$ and subtracting the result from the dividend produces:

Dividing $x^{2}$ by $x$ yields $x$. Multiplying and subtracting we obtain:

Dividing $3 x$ by $x$ yields $3$. Multiplying and subtracting:

The remainder is $1$, confirming that $R = P ( 2 ) = 1$ in accordance with the remainder theorem. The quotient and remainder are:

The division can be written as:

Rational functions and polynomial division

When the division of two polynomials is performed without separating the remainder, the result is represented as a rational function where $D ( x ) \neq 0$:

In this context, polynomial division offers a systematic method to decompose $F ( x )$ into a polynomial component and a proper rational component. A proper rational function is defined as one in which the numerator has a strictly lower degree than the denominator:

This decomposition is unique. The polynomial part $Q ( x )$ and the proper rational part $R ( x ) / D ( x )$ are uniquely determined by $P ( x )$ and $D ( x )$, as a direct consequence of the uniqueness of the division algorithm.

This decomposition serves as the foundation for partial fraction decomposition, a technique that expresses the proper rational component as a sum of simpler fractions. This method is widely utilised in integration.

Selected References

• Cornell University, J. Belk. Number Theory for Polynomials
• University of Connecticut, K. Conrad. The Division Algorithm in Z
• Harvard University, S. Vadhan. Polynomial Rings and Division with Remainder
• Harvard University, C. McMullen. Algebra II: Rings and Fields