Question

Show that (2x + 1) is a factor of 4x³ + 12x² + 11 x + 3 .Hence factorise 4x³ + 12x² + 11x + 3.

Answer 1

Let 2x + 1 = 0,
then x = 
Substituting the value of x in f(x),
f(x) = 4x³ + 12x² + 11x + 3

`f (-1/2) = 4(-1/2)^3 + 12(-1/2)^2 + 11(-1/2) + 3`

= `4(-1/8) + 12(1/4) + 11(-1/2) + 3`

= `4(1)/(2) + 3 - (11)/(2) + 3`
= (6) – (6)
= 0
∵ Remainder = 0
∴ 2x + 1 is a factor of
4x³ + 12x² + 11x + 3
Now dividing f(x) by 2x + 1, we get

`2x + 1")"overline(4x^3 + 12x^2 + 11x + 3)("2x^2 + 5x + 3`
           4x³ +  2x²              
            –    –                    
                  10x²  + 11x
                  10x²  + 5x
                   –       –           
                           6x + 3
                           6x + 3
                            –   –       
                                x        
∴ 4x³ + 12x² + 11x + 3
= (2x + 1)(2x² + 5x + 3)
= (2x + 1)[2x² + 2x + 3x + 3]
= (2x + 1)[2x(x + 1) + 3(x + 1]
= (2x + 1)[(x + 1)(2x + 3)]
= (2x + 1)(x + 1)(2x + 3).