Polynomial Division

By using a method similar to dividing a number, we can also divide the polynomials, for example (3x3 – 7x2 – 11x + 4) : (x – 4) in the following distribution model,

Information that must be known:
(i) x - 4 as a divider
(ii) 3x2 + 5x + 9  is the result of the division of the polynomial
(iii) 40 is the remainder of the division

Look again at the division model above. If the terms (symbols of numbers) printed in italics we delete them, then the following simpler form will be obtained,
If the top row is not written down and only the coefficient is observed, the following form is obtained,
Description: arrows indicate multiply -4.
The process can be simplified by changing -4 to 4, and the subtraction operation is changed to addition.

By paying close attention to the scheme, it can be explained that
• The remaining share is f(4), which in this case is 40.
• The third line coefficient before f(4) is the yield coefficient, in this case 3x2 + 5x + 9.

The above division process is called the process of sharing Synthetic or division by means of Horner.

Example 1.
Determine the quotient and the remainder of 2x3 – 4x2 – 5  with x + 3.

The results of the distribution are 2x2 – 10x + 30 and the remainder is -95