Polynomial division rewrites one polynomial divided by another as a quotient plus a possible remainder.
It is similar to ordinary number division, but the terms are powers of a variable.
1. The polynomial division identity
If \(P(x)\) is divided by \(D(x)\), then:
\[
\begin{aligned}
P(x)=D(x)Q(x)+R(x),
\end{aligned}
\]
where \(Q(x)\) is the quotient and \(R(x)\) is the remainder.
The remainder must have degree smaller than the divisor:
\[
\begin{aligned}
\deg R(x)<\deg D(x).
\end{aligned}
\]
2. Polynomial long division
Polynomial long division works for any polynomial divisor.
The main idea is to divide the leading term of the current remainder by the leading term of the divisor.
For example, divide:
\[
\begin{aligned}
x^3+5x^2-2x-8
\quad\text{by}\quad
x+2.
\end{aligned}
\]
First divide leading terms:
\[
\begin{aligned}
\frac{x^3}{x}=x^2.
\end{aligned}
\]
Multiply and subtract:
\[
\begin{aligned}
x^3+5x^2-2x-8
-
(x^2)(x+2)
&=
x^3+5x^2-2x-8-(x^3+2x^2)\\
&=
3x^2-2x-8.
\end{aligned}
\]
Repeat:
\[
\begin{aligned}
\frac{3x^2}{x}=3x.
\end{aligned}
\]
\[
\begin{aligned}
3x^2-2x-8
-
3x(x+2)
&=
3x^2-2x-8-(3x^2+6x)\\
&=
-8x-8.
\end{aligned}
\]
Repeat once more:
\[
\begin{aligned}
\frac{-8x}{x}=-8.
\end{aligned}
\]
\[
\begin{aligned}
-8x-8
-
(-8)(x+2)
&=
-8x-8-(-8x-16)\\
&=
8.
\end{aligned}
\]
So:
\[
\begin{aligned}
\boxed{
x^3+5x^2-2x-8
=
(x+2)(x^2+3x-8)+8
}.
\end{aligned}
\]
3. Synthetic division
Synthetic division is a shortcut for dividing by a monic linear divisor:
\[
\begin{aligned}
x-r.
\end{aligned}
\]
If the divisor is \(x+2\), rewrite it as:
\[
\begin{aligned}
x+2=x-(-2).
\end{aligned}
\]
Therefore the synthetic root is:
\[
\begin{aligned}
r=-2.
\end{aligned}
\]
4. Synthetic division example
Divide:
\[
\begin{aligned}
x^3+5x^2-2x-8
\quad\text{by}\quad
x+2.
\end{aligned}
\]
Use the coefficients:
\[
\begin{aligned}
1,\quad 5,\quad -2,\quad -8.
\end{aligned}
\]
The synthetic root is \(-2\). Bring down the first coefficient:
\[
\begin{aligned}
1.
\end{aligned}
\]
Multiply by \(-2\), add to the next coefficient, and continue:
\[
\begin{aligned}
1,\quad 3,\quad -8,\quad 8.
\end{aligned}
\]
The quotient coefficients are \(1,3,-8\), and the remainder is \(8\).
Therefore:
\[
\begin{aligned}
\boxed{
\frac{x^3+5x^2-2x-8}{x+2}
=
x^2+3x-8+\frac{8}{x+2}
}.
\end{aligned}
\]
5. Why synthetic division works
Synthetic division is a compact version of long division.
It keeps only the coefficients because the divisor \(x-r\) has a simple leading coefficient of \(1\).
The multiplication-and-addition pattern comes from repeatedly subtracting multiples of \(x-r\).
6. Remainder theorem
If a polynomial \(P(x)\) is divided by \(x-r\), the remainder is:
\[
\begin{aligned}
P(r).
\end{aligned}
\]
This is called the remainder theorem.
For example, when dividing by \(x+2\), use \(r=-2\):
\[
\begin{aligned}
P(-2)=8.
\end{aligned}
\]
That matches the synthetic division remainder.
7. Factor theorem
If the remainder is zero, then \(x-r\) is a factor of \(P(x)\):
\[
\begin{aligned}
P(r)=0
\quad\Longleftrightarrow\quad
x-r\text{ is a factor of }P(x).
\end{aligned}
\]
This is useful for factoring polynomials.
8. When to use each method
Use synthetic division only when the divisor has the form \(x-r\), such as \(x-3\) or \(x+2\).
Use long division for more general divisors, such as \(2x+1\), \(x^2-1\), or \(x^2+3x+5\).
9. Missing powers
In synthetic division, every power must have a coefficient.
If a term is missing, its coefficient is \(0\).
For example:
\[
\begin{aligned}
x^3-8
\end{aligned}
\]
should be written with coefficients:
\[
\begin{aligned}
1,\quad 0,\quad 0,\quad -8.
\end{aligned}
\]
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Using synthetic division for a divisor that is not of the form \(x-r\).
- Using \(2\) instead of \(-2\) when the divisor is \(x+2\).
- Forgetting zero coefficients for missing powers.
- Subtracting incorrectly during long division.
- Stopping before the remainder has smaller degree than the divisor.
- Forgetting to write the final answer as quotient plus remainder over divisor.
Key idea: synthetic division is a shortcut for \(x-r\); long division is the general method for polynomial division.
