Find all the roots of fx 2x3 x2 12x 9 given 3 as a root

Find all the roots of f(x) = 2x^3 - x^2 - 12x - 9 given 3 as a root. Write the function in factored form.

Solution

given 3 is a root

=>(x-3) is a factor of f(x)

long division:

x-3......]....2x³-x²-12x-9....[...2x²+5x+3
................2x³-6x²...................
.....................5x²-12x..............
.....................5x²-15x..............
............................3x-9............
............................3x-9............
...............................0..............

so f(x)=(x-3)(2x²+5x+3)

for roots f(x)=0

(x-3)(2x²+5x+3) =0

=>(2x²+5x+3)=0

=>2x²+5x= -3

=>x²+(5_/2)x=-(3_/2)

=>x²+(5_/2)x+(5_/4)²=-(3_/2)+(5_/4)²

=>(x+(5_/4))²=-(3_/2)+(25_/16)

=>(x+(5_/4))²=(1_/16)

=>(x+(5_/4))=-(1_/4) ,(x+(5_/4))=(1_/4)

=>x=-(5_/4)-(1_/4) ,x=-(5_/4)+(1_/4)

=>x=(-3_/2) ,x=-1

roots of f(x) are x=(-3_/2),-1,3

=>2x+3 , x+1 ,x-3 are factors of f(x)

so f(x)=2x³-x²-12x-9 =(2x+3)(x+1)(x-3)