Today I learned that Horner method not only computes the value of a polynomial at a given point but also gives the ratio and the remainder of and .
Let . We can guess that is a root. (See below about how to make an educated guess.) Dividing by gives . (Instead of describing this division, I'll describe the next one.) Again by guessing, is a root of .
To divide by , make a table with three rows. The first row consists of the coefficients of . The first element of the second row is 0. The third row is the sum of the first two. Therefore, the first element of the third row is 2.
The second and each subsequent element of the second row is times the element of the third row immediately to the left. So, 4 = 2 * 2, -6 = 2 * (-3), and -18 = 2 * (-9). One computes the second and third row simultaneously left to right. The result indicates that the ratio is the polynomial and the remainder is 0. Now, is a quadratic equation.
2 -7 -3 18
0 4 -6 -18
2 -3 -9 0
To make an educated guess about the roots, one can use the rational root theorem. It says that if a rational number (in the lowest terms) is a root of , then divides the constant term of and divides the leading coefficient. So, for , all rational roots are among . Note that the theorem says nothing about irrational roots.