Step 1

Zeros of a polynomial:

\(p(x)=x^{4}+6x^{3}+6x^{2}-18x-27\)

To find zeros of a polynomial we need to put p(x)=0 and then simplify the polynomial in factor form.

Step 2

\(x^{4}+6x^{3}+6x^{2}-18x-27=0\)

\(x^{4}+3x^{3}+3x^{3}+9x^{2}-3x^{2}-9x-9x-27=0\)

\(x^{3}(x+3)+3x^{2}(x+3)-3x(x+3)-9(x+3)=0\)

\((x+3)(x^{3}+3x^{2}-3x-9)=0\)

\((x+3)(x^{2}(x+3)-3(x+3))=0\)

\((x+3)(x+3)(x^{2}-3)=0\)

\((x+3)(x-3)(x+\sqrt{3})(x-\sqrt{3})=0\)

The zeros are (x+3)=0. x=-3

(x-3)=0. x=3

\((x+\sqrt{3})=0. x=-\sqrt{3}\)

\((x-\sqrt{3})=0. x=\sqrt{3}\)

The zeros are \(x = 3,-3,\sqrt{3},-\sqrt{3}\)

=3,-3,1.7,-1.7

The values of A,B,C,D are 3,-3,-3,-9 respectively.

Zeros of a polynomial:

\(p(x)=x^{4}+6x^{3}+6x^{2}-18x-27\)

To find zeros of a polynomial we need to put p(x)=0 and then simplify the polynomial in factor form.

Step 2

\(x^{4}+6x^{3}+6x^{2}-18x-27=0\)

\(x^{4}+3x^{3}+3x^{3}+9x^{2}-3x^{2}-9x-9x-27=0\)

\(x^{3}(x+3)+3x^{2}(x+3)-3x(x+3)-9(x+3)=0\)

\((x+3)(x^{3}+3x^{2}-3x-9)=0\)

\((x+3)(x^{2}(x+3)-3(x+3))=0\)

\((x+3)(x+3)(x^{2}-3)=0\)

\((x+3)(x-3)(x+\sqrt{3})(x-\sqrt{3})=0\)

The zeros are (x+3)=0. x=-3

(x-3)=0. x=3

\((x+\sqrt{3})=0. x=-\sqrt{3}\)

\((x-\sqrt{3})=0. x=\sqrt{3}\)

The zeros are \(x = 3,-3,\sqrt{3},-\sqrt{3}\)

=3,-3,1.7,-1.7

The values of A,B,C,D are 3,-3,-3,-9 respectively.