QUESTION 1
The given function is
[tex]f(x)=\frac{(x-3)(x-1)}{(x-1)(x+2)}[/tex]
We simplify to get;
[tex]f(x)=\frac{x-3}{x+2}[/tex]
This function is equal to zero when [tex]x-3=0[/tex]
[tex]\Rightarrow x=3[/tex]
QUESTION 2
The given function is
[tex]f(x)=\frac{5x^2-10x+5}{2x^2-5x+3}[/tex]
[tex]f(x)=\frac{5(x^2-2x+1)}{2x^2-5x+3}[/tex]
We factor to get;
[tex]f(x)=\frac{5(x-1)^2}{(2x-3)(x-1)}[/tex]
[tex]f(x)=\frac{5(x-1)}{(2x-3)}[/tex]
This function equals zero when
[tex]5(x-1)=0[/tex]
[tex]x=1[/tex]
QUESTION 3
The given function is
[tex]f(x)=\frac{x^3-8}{x^2-6x+8}[/tex]
We factor to get,
[tex]f(x)=\frac{(x-2)(x^2+2x+4)}{(x-2)(x+4)}[/tex]
[tex]f(x)=\frac{x^2+2x+4}{x+4}[/tex]
The function equals zero when [tex]x^2+2x+4=0[/tex]
[tex]D=b^2-4ac[/tex]
[tex]D=2^2-4(1)(4)[/tex]
[tex]D=-12[/tex]
Hence the equation has no real roots.
The complex roots are
[tex]x=-\sqrt{3}i-1[/tex] or [tex]x=-1+\sqrt{3}i[/tex]
