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
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) =`
Therefore the roots of the equation
`4x^3 +12x^2 -x -3 =0`
are the roots of the equivalent equation
`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.