Synthetic Division

When dividing polynomials of the form p(x)/(x-a) we can use synthetic division as a shortcut for polynomial long division. Below we divide using traditional polynomial long division and synthetic division side by side.

Both processes give the same result, x^2 - 3x - 2. However, synthetic division uses only the coefficients and requires much less writing.  To understand synthetic division, we walk you through the process below. Be sure the polynomials are in standard form, that is, each term is arranged in descending order from highest power to lowest.

Step 1: Write the root a determined from (x-a) and the coefficients of the polynomial in the first line.
Step 2: Bring down the first coefficient and we are ready to begin.
Step 3: Multiply a by the first coefficient and write the result under the second coefficient.
Step 4: Add the second column and write the result below.
Step 5: Repeat the process for all of the remaining columns.
Step 6: The numbers along the bottom are the coefficients of the result in standard form beginning with a term of degree one less than the original polynomial. The last number is the remainder.
To finish, clearly present the answer to your reader.  Next we do an example with a remainder. Just as we do in polynomial long division, we add a term that is the remainder over the divisor.

Here the root (or zero) of (x+5)  is -5.
Multiplying both sides by the divisor (x+5) we have the following.
We mentioned that the polynomials are required to be in standard form. Sometimes there will be "missing terms."  That is, not all powers will have nonzero coefficients.  In this case, we use 0 as placeholders when performing synthetic division.


Sometimes the root will be a fraction.


It is interesting to note that the result has a GCF of 2 and we can do the following algebraic manipulations:
In short, this gives us a method of factoring a more complicated polynomial.