Question

Factorize completely using factor theorem:

2x³ – x² – 13x – 6

Answer 1

Let f(x) = 2x³ – x² – 13x – 6

By hit and trial method

Put x = – 2

f(–2) = 2(–2)³ – (–2)² – 13(–2) – 6

= – 16 – 4 + 26 – 6

= 0

${\displaystyle \Rightarrow }$ (x + 2) is a factor of f(x)

∴ Dividing f(x) by (x + 2)

${\displaystyle x+2\text{)}\overline{2x^3-x^2-13x-6}(2x^2-5x-3}$
             2x³ + 4x²
              –    –                       
                    –5x² – 13x – 6
                    –5x² – 10x
                   +        +             
                             –3x – 6
                             –3x – 6
                      +       +           
                           x                

So, 2x³ – x² – 13x – 6 = (x + 2)(2x² – 5x – 3)

= (x + 2){2x² – 6x + x – 3}

= (x + 2){2x (x – 3) + 1(x – 3)}

= (x + 2)(x – 3)(2x + 1)