Synthetic division is a compact way to divide a polynomial by a linear expression of the form \(x-c\). It uses the polynomial’s coefficients instead of writing every power of x at each step. The result gives a quotient and a remainder.
Start with a polynomial
A polynomial is an expression such as \(2x^3-5x^2-4x+3\). Its variable exponents are nonnegative integers, and its coefficients may be positive, negative, zero, or noninteger real numbers. A polynomial expression becomes part of an equation when it is set equal to something.
The standard synthetic division procedure below applies to division by \(x-c\). For other divisors, polynomial long division is a more general method.
Worked example: divide by x − 3
Divide \(2x^3-5x^2-4x+3\) by \(x-3\).
- Use 3. Write the divisor as \(x-c\); here, \(c=3\).
- List the coefficients: \(2,-5,-4,3\), in descending order of powers.
- Bring down 2. This is the first coefficient of the quotient.
- Multiply and add: \(3\times2=6\), then \(-5+6=1\).
- Repeat: \(3\times1=3\), then \(-4+3=-1\).
- Finish: \(3\times(-1)=-3\), then \(3-3=0\).
\[\begin{array}{r|rrrr}3&2&-5&-4&3\\ &&6&3&-3\\\hline &2&1&-1&0\end{array}\]
The last entry is the remainder. The other entries are the coefficients of a polynomial one degree lower than the original:
\[\text{Quotient}=2x^2+x-1,\qquad\text{remainder}=0.\]
Thus,
\[2x^3-5x^2-4x+3=(x-3)(2x^2+x-1).\]
Expanding the right-hand side verifies every coefficient. Since the remainder is zero, \(x-3\) is a factor and \(x=3\) is a root of the original polynomial.
Missing powers need zero coefficients
Now divide \(x^3-4x+2\) by \(x+2\). Rewrite the divisor as \(x-(-2)\), so use \(-2\). The coefficient list is \(1,0,-4,2\): the zero represents the missing \(x^2\) term.
Bring down 1. Multiplying by \(-2\) and adding successively gives \(-2\), then 0, then 2. The quotient is \(x^2-2x\), and the remainder is 2:
\[x^3-4x+2=(x+2)(x^2-2x)+2.\]
For \(x\ne-2\), the corresponding division can be written as
\[\frac{x^3-4x+2}{x+2}=x^2-2x+\frac2{x+2}.\]
The polynomial identity remains true at \(x=-2\), but the fraction is undefined there because its denominator is zero. There is no general restriction that x must be nonzero.
The remainder theorem
When a polynomial \(P(x)\) is divided by \(x-c\), its remainder is \(P(c)\). In the second example,
\[P(-2)=(-2)^3-4(-2)+2=2,\]
which agrees with the synthetic division result. A remainder of zero means \(x-c\) is a factor; a nonzero remainder means it is not.
Common mistakes
- Using the wrong sign: divide by \(x+2\) using \(-2\), not 2.
- Omitting zero placeholders for missing powers.
- Reading the final remainder as another quotient coefficient.
- Applying the basic procedure unchanged to a divisor such as \(2x-3\).
Continue with factoring or the quadratic formula when the quotient needs further analysis.
Further reading
OpenStax covers both polynomial long division and synthetic division.