Question

Factorise
$2 {x}^{3}-3 {x}^{2} - 17x + 30$

Answer 1

2x³ - 3x² - 17x + 30 = 

note that x = 2 is a zero of the polynomial: 

2(2)³ - 3(2)² - 17(2) + 30 = 

2(8) - 3(4) - 34 + 30 = 

16 - 12 - 4 = 0 

then the polynomial is divisible by (x - 2): 

let's verify this by synthetic division: 

...2 | 2...- 3.....-17....+30 
......| 
......| .....+4.....+2.....-30 
----------------------------------- 
.......2....+1.....-15.......0 


we get: 

(2x³ - 3x² - 17x + 30) /(x - 2) = 2x² + (1)x - 15 

therefore: 

(2x³ - 3x² - 17x + 30) = (x - 2)(2x² + x - 15) 

let's now factor the quadratic replacing x with 6x - 5x: 

2x² + x - 15 = 2x² + 6x - 5x - 15 = 

(factoring by grouping) 

2x(x + 3) - 5(x + 3) = 

(factoring the common term (x + 3)) 

(x + 3)(2x - 5) 

thus the answer is: 


(2x³ - 3x² - 17x + 30) = (x - 2)(x + 3)(2x - 5)