Solve the given polynomial equation use the ration zero theo

Solve the given polynomial equation use the ration zero theorem and Descartes\'s rule of signs as an did in obtain the first root 4x^3 - 12x^2 - 9x - 1 = 0 the solution set is use commas to separate answer type of fraction type exact answer using radicals as needed.

Solution

4x^3 -12x^2 -9x -1 =0

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction p/q, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factors of the leading coefficient (4) are 1 2 2 4 .The factor of the constant term (-1) is 1 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±1/2, ±1/2, ±1/4

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial P(x), we obtain P(1/2)=0.

(4x^3 -12x^2 -9x -1)/(2x +1)

= 2x^2 - 7x -1

solve the quadratic we get x= 7/4 +/-sqrt(57)/4

So, roots of polynomial are : x= -1/2 ,  7/4 +/-sqrt(57)/4