Question

Use factor theorem to factorise 6x³ + 17x² + 4x − 12.

Answer 1

Let f(x) = 6x³ + 17x² + 4x − 12

Factors of constant term −12 are ±1, ±2, ±3, ±4, ±6

Put x = 1;

f(1) = 6(1)³ + 17(1)² + 4(1) − 12

= 6 + 17 + 4 − 12

= 15

So (x – 1) is not a factor.

Put x = –1;

f(–1) = 6(–1)³ + 17(–1)² + 4(–1) − 12

= –6 + 17 – 4 – 12

= –5

So (x + 1) is not a factor.

Put x = 2;

f(2) = 6(2)³ + 17(2)² + 4(2) − 12

= 6(8) + 17(4) + 4(2) − 12

= 48 + 68 + 8 − 12

= 112

So (x – 2) is not a factor.

Put x = –2;

f(–2) = 6(–2)³ + 17(–2)² + 4(–2) − 12

= 6(–8) + 17 (–4) + 4(–2) − 12

= –48 + 68 – 8 – 12

= 0

So x + 2 is a factor of p(x).

            6x² + 5x – 6
`x + 2")"overline(6x^3 + 17x^2 + 4x - 12)`
           6x³ + 12x²
         −      −                             
                      5x² + 4x
                      5x² + 10x
                   −       −                  
                              – 6x – 12
                              – 6x – 12
                              +      +       
                                     x         

6x³ + 17x² + 4x − 12 = (x + 2) (6x² + 5x – 6)

= (x + 2) (6x² + 9x – 4x – 6)

= (x + 2) [3x(2x + 3) – 2(2x + 3]

= (x + 2) (3x – 2) (2x + 3)