Synthetic Division

second degree polynomial, synthetic division, irreducible polynomials, coefficients, remainder

Synthetic Divsion is a way to find the factorization of a given polynomial equation after we figure out what possible factors we can try.  Thus, given a polynomial function, if we think that it has a certain factor, say 2, as one of the irreducible factors, then, we use synthetic division to see if (x-2) is a factor of the polynomial function and reduce the polynomial to a degree that is one lower than we started out with.  We continue the process until the polynomial can not be reduced any further.  

Note that when we use synthetic division, we try the root 2 and not the factor we guess, (x-2).  

Example:

Suppose we are given a polynomial equation like: x³3+2x²-9x-18.  Synthetic division is as follows:

We write down the coefficient of the polynomial function, keep the factor we think will work on the outside and proceed through multiplication and adding methods.  The factors we want to guess have to be the factors of our constant term divided by the leading coefficient.  Here, it is -18/1.  So, possible factors are +-1, +-2, +-3, +-6, +-9,+-18.  So, of these, suppose we want to try "-2".  Then, we do the following:

    -2 |  1  2  -9   -18
        |
        |___________

We bring down the 1, the leading coefficient. We then multiply by the factor "-2". Next we add this value, -2, to the next coefficient and write the sum below the bottom line. We then repeat this process for every coefficient.

-2 |  1  2  -9   -18
     |    -2   0  +18
     |____________
        1  0  -9     0

We see that after the process is done, we have a remainder of "0".  This is great because it means that (x+2) was a factor of our original function.  More importantly, the numbers that we have at the bottom are the coefficients of the reduced polynomial which together with our factor gives us the original polynomial.  Thus, from the numbers at the bottom, we can gather that the reduced polynomial is 1x²+0x-9, or x²-9.

So, we get that  (x^3+2x^2-9x-18)= (x+2)(x^2-9)

We can continue this process until we have reduced our polynomial to products of irreducible polynomials.  However, we can also take note of the fact that the second degree polynomial we have is just the difference of two squares so we can conclude that:

(x^3+2x^2-9x-18)= (x+2)(x^2-9)=(x+2)(x-3)(x+3).

The important thing to remember with synthetic division is it is only useful if we have an idea about what our possible roots are. Without an idea synthetic division is very slow and difficult to find a relevant value.

second degree polynomial, synthetic division, irreducible polynomials, coefficients, remainder