Find all zeros of the polynomial 6x4+17x3-2x2+x-6 = 0

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We can write the given polynomial 6*x^4+17*x^3-2*x^2+x-6 as:

6*x^4+17*x^3-2*x^2+x-6

=> 6x^4 + 18x^3 - x^3 - 3x^2 + x^2 + 3x - 2x - 6

=> 6x^3( x + 3) - x^2(x + 3) + x(x + 3) - 2(x + 3)

=> (x + 3)[6x^3 - x^2 + x - 2]

=> (x + 3)( 6x^3 - 4x^2 + 3x^2 - 2x + 3x - 2)

=> (x +3)(2x^2(3x - 2) + x(3x - 2) + 1(3x - 2))

=> (x +3)(3x - 2)( 2x^2 + x +1)

Equating the polynomial to 0

(x +3)(3x - 2)( 2x^2 + x +1)

x + 3 = 0

=> x1 = -3

3x - 2 = 0

=> x2 = 2/3

2x^2 + x +1 = 0

=> x3 = [-1 + sqrt (1 - 8)]/4

=> x3 = -1/4 + (i*sqrt 7) / 4

x4 = -1/4 - (i*sqrt 7) / 4

Therefore the roots or zeroes of the polynomial 6*x^4+17*x^3-2*x^2+x-6 are :

-3 , 2/3 , [(-1/4) + (i*sqrt 7)/4] , [(-1/4) - (i*sqrt 7)/4]

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We can find the zeros of a polynomial equation by using synthetic division.

First check to see that the equation is written in descending powers of the variable.

Then look at the highest power in the equation.  This will tell how many solutions (zeros) there are for the equation.

6x^4   +  17x^3  -  2x^2  +  x  -  6   =   0

The degree of the equation is 4; therefore, we will have 4 solutions.

Synthetic Division:

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