Question

How do you simplify `(3x^2-5x-2)/(6x^3+2x^2+3x+1)`?

Answer 1

`color(red)((3x^2-5x-2))/color(purple)((6x^3+2x^2+3x+1))`

• Factorising the numerator :
  By splitting the middle term
  `color(red)(3x^2-5x-2`

`3x^2-5x-2 = 3x^2-6x + 1x -2`
`= 3x(x-2)+1(x-2)`
`= color(red)((3x+1)(x-2)`

• Factorising the denominator:
  By grouping
  `color(purple)(6x^3+2x^2+3x+1)`

`= (6x^3+2x^2)+(3x+1)`
`= 2x^2(3x +1)+1(3x+1)`
`= color(purple)((2x^2+1) (3x +1)`

The expression becomes:
`= color(red)(cancel(3x+1)(x-2))/ color(purple)((2x^2+1) cancel(3x +1)`
`= color(red)((x-2))/ color(purple)((2x^2+1)`