# Divisibility

Definition:[br][br] [color=#0000ff] A polynomial expression P(x) is said to be divisible if there exist polynomial expressions Q(x) and R(x) so that[br][br] P(x) = Q(x)R(x),[br][br]where 1 $\large \textcolor{blue}{\leq}$ degree of Q(x) and R(x) < degree of P(x).[/color][br][br]Linear polynomial expression cannot be factorized according to the definition.[br][br]In general,[br][br] ﻿ ﻿ ﻿ $\Large P(x) = a_n x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0 \\[br]\Large = a_n(x-x_1)(x-x_2)\cdots(x-x_n)$[br][br]where $\Large x_1 ,x_2, \ldots , x_n$ are zeros of the polynomial and $\Large n \geq 1,a_n\neq0$ .[br][br]All quadratic polynomial expressions can be factorized by finding the zeros and using the generalization given above.[br][br] [br][color=#0000ff][br][u]Example 1.[/u][/color] Factorize $\Large 36x^2-3x-18$[br][br]By using the formula of quadratic equations, we get the zeros of this expression: [br] [br][br] ﻿ [br] ﻿ $\Large 36x^2-3x-18=0\;\;\Leftrightarrow\;\;x=\frac{3}{4}\;\vee\;x=-\frac{2}{3}.$ ﻿ [br][br]Thus,[br][br] ﻿ ﻿$\Large 36x^2-3x-18=36(x-\frac{3}{4})[x-(-\frac{2}{3})] \\ =\Large 3\cdot3 \cdot4 (x-\frac{3}{4} )(x+\frac{2}{3})=3(4x-3)(3x+2)$ ﻿[br][br][br][br]