Question

Whats the roots of the following three problems:

1.
`F(x) = ((x-3)(x-1))/((x-1)(x+2))`
2.
`F(x)(5x^(2)-10x+5 )/(2x^(2)-5x +3)`
3.
`F(x) = (x^(3)-8 )/(x^(2)-6x+8 )`

Answer 1

QUESTION 1

The given function is

`f(x)=((x-3)(x-1))/((x-1)(x+2))`

We simplify to get;

`f(x)=(x-3)/(x+2)`

This function is equal to zero when
`x-3=0`

`\Rightarrow x=3`

QUESTION 2

The given function is

`f(x)=(5x^2-10x+5)/(2x^2-5x+3)`

`f(x)=(5(x^2-2x+1))/(2x^2-5x+3)`

We factor to get;

`f(x)=(5(x-1)^2)/((2x-3)(x-1))`

`f(x)=(5(x-1))/((2x-3))`

This function equals zero when

`5(x-1)=0`

`x=1`

QUESTION 3

The given function is

`f(x)=(x^3-8)/(x^2-6x+8)`

We factor to get,

`f(x)=((x-2)(x^2+2x+4))/((x-2)(x+4))`

`f(x)=(x^2+2x+4)/(x+4)`

The function equals zero when
`x^2+2x+4=0`

`D=b^2-4ac`

`D=2^2-4(1)(4)`

`D=-12`

Hence the equation has no real roots.

The complex roots are

`x=-√(3)i-1` or
`x=-1+√(3)i`