Question

What does the fundamental theorem of algebra illustrate?

Answer 1

The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.

Explanation:

The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.

We have to find the roots of this given equation.

If a quadratic equation is of the form
`ax^(2)+bx +c=0`

Its roots are
`\frac{-b+\sqrt{b^(2)-4ac } }{2a}` and
`\frac{-b-\sqrt{b^(2)-4ac } }{2a}`

Here the given equation is
`2x^(2)-4x-1` = 0

a = 2

b = -4

c = -1

If the roots are
`x_(1) and x_(2)`, then

`x_(1)` =
`\frac{-2+\sqrt{(-4)^(2)-4* 2* (-1)}}{2* 2}`

=
`(4 +√(24))/(4)`

=
`(2+√(6) )/(2)`

`x_(2)` =
`\frac{-2-\sqrt{(-4)^(2)-4* 2* (-1)}}{2* 2}`

=
`(4 +√(8))/(4)`

=
`(2-√(6) )/(2)`

These are the two roots of the equation.