Answer: The correct option is 2.
Explanation:
The given polynomial is,
[tex]f(x)=x^3-x^2+6x-6[/tex]
By rational root theorem 1 and -1 are possible roots of all polynomials.
put x=1
[tex]f(x)=(1)^3-(1)^2+6(1)-6=0[/tex]
Since the value of f(x) is 0 at x=1. So 1 is a real root of the polynomial.
Divide the given polynomial by (x-1) using synthetic division method.
[tex]f(x)=(x-1)(x^2+6)[/tex]
Equate each factor equal to 0.
[tex]x=1[/tex]
[tex]x=\pm\sqrt{-6}[/tex]
So, the given polynomial have one real root and 2 non real roots. Since the function have one real root therefore the function intersects the x-axis at exactly one location.
Thus, the correct option is 2.
