Synthetic Methods

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.

Let's look at the quadratic equation: y = x2 + 5x + 6. From the rational root testwe  know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring, we know that the zeroes are, in fact, –3 and –2.) How would we use synthetic division to check the potential zeroes?  If we guess that x = 1 is a zero, then this means that x – 1 is a factor of the quadratic. And if it's a factor, then it will divide out evenly; that is, if we divide x2 + 5x + 6 by x – 1, we would get a zero remainder.

we divide x2 + 5x + 6 by x – 1 by synthetic method as follows:

1. write the coefficients ONLY inside an upside-down division symbol:

write coefficients in upside-down division symbol

Make sure you leave room inside, underneath the row of coefficients, to write another row of numbers later.


2.Put the test zero, x = 1, at the left:

write test zero at left

3. Take the first number inside, representing the leading coefficient, and carry it down, unchanged, to below the division symbol:

carry down leading coefficient


4.Multiply this carry-down value by the test zero, and carry the result up into the next column:

multiply by test zero, and carry result up into next column


5.Add down the column:

add down the column

6. Multiply the previous carry-down value by the test zero, and carry the new result up into the last column:

multiply result by test zero, and carry result into next column


7.Add down the column:

This last carry-down value is the remainder.

that means (x2 + 5x + 6) ÷ (x – 1) =12

Hence x-1 is not a factor of the polynomial x2 + 5x + 6