Question

Find all the zeros of the equationX^4-6x^2-7x-6=0

Answer 1

Consider the following equation:

`x^4-6x^2-7x-6=0`

First, to solve this equation, we must factor it:

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

using the zero factor theorem, the following must be met:

Equation 1:

`(x+2)=0`

or

Equation 2:

`(x-3)=0`

or

Equation 3:

`(x^2+x+1)=0`

From equation 1, solving for x, we obtain:

`x\text{ = -2}`

or from equation 2, solving for x, we obtain:

`x\text{ = 3}`

Now, remember the following quadratic formula to find the solutions of a quadratic equation:

`\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

applying this to equation 3, where

a = 1

b= 1

and

c = 1

we obtain:

`\frac{-1\pm\sqrt[]{1-4}}{2}=\frac{-1\pm\sqrt[]{-3}}{2}=\frac{-1\pm\sqrt[]{3}i\text{ }}{2}=-(1)/(2)\pm\frac{\sqrt[]{3}i\text{ }}{2}`

then, we obtain two additional solutions:

`-(1)/(2)+\frac{\sqrt[]{3}i\text{ }}{2}`

or

`-(1)/(2)-\frac{\sqrt[]{3}i\text{ }}{2}`

so that, we can conclude that the correct answer is:

The solutions (zeros) for the given equation are:

`x\text{ = -2}`

`x\text{ = 3}`

`-(1)/(2)+\frac{\sqrt[]{3}i\text{ }}{2}`

`-(1)/(2)-\frac{\sqrt[]{3}i\text{ }}{2}`