Divisibility
- Author:
- P Porras
Definition:
A polynomial expression P(x) is said to be divisible if there exist polynomial expressions Q(x) and R(x) so that
P(x) = Q(x)R(x),
where 1 degree of Q(x) and R(x) < degree of P(x).
Linear polynomial expression cannot be factorized according to the definition.
In general,
where are zeros of the polynomial and .
All quadratic polynomial expressions can be factorized by finding the zeros and using the generalization given above.
Example 1. Factorize
By using the formula of quadratic equations, we get the zeros of this expression:
Thus,
Define a so that polynomial is divisible with
If is divisible with , they must have a common zero. As then polynomial must equal to zero, if By substituting
Simplify
Factorise both a numerator and a denominator, and simplify common terms: