Find every combination of `+-p/q` These are the possible roots of the polynomial function.

`+-1, +- 1/2,+- 1/4,+-3,+-3/2,+-3/4`

Since `1/2` is a known root, divide the polynomial by `(x - 1/2)` to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

`(4x^3+ 12x^2 - x - 3)/(x - 1/2)`

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by `1` .

`4x^2 + 14x + 6`

The polynomial can be written as a set of linear factors.

`(x - 1/2) (x + 3) (x + 1/2 )`

These are the roots (i.e. zeros) of the polynomial `4x^3 + 12x^2 - x - 3`

`x = 1/2 , - 3 , -1/2`

*A graph is attached below*

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To factorize the above expression we group two by two the terms

`4x^3 +12x^2 -x -3 =4x^2(x+3) -(x+3) =(4x^2-1)(x+3) =`

`=(2x+1)(2x-1)(x+3)`

Therefore the roots of the equation

`4x^3 +12x^2 -x -3 =0`

are the roots of the equivalent equation

`(2x+1)(2x-1)(x+3) =0`

`x_1 =-1/2` , `x_2 =1/2` , `x_3 =-3`

The graph of the function `f(x) =4x^3 +12x^2 -x -3` is attached below.