Answer:
All roots of the given function are, 2 , 3 and 4
Step-by-step explanation:
Given the function f(x) = [tex]x^3-9x^2+26x-24[/tex]
Also, it given that x=2 is a root of the function f(x).
So, (x-2) is a factor of f(x).
Remainder theorem states that if a function is divided by (x-a), then the remainder is equal to f(a). If a function f(a) is equal to 0, therefore a is the root of the function.
We use synthetic method to divide f(x) by (x-2) as also shown in figure below;
[tex]f(x) = (x-2)(x^2-7x+12)[/tex]
[tex]f(x)=(x-2)(x^2-3x-4x-12)[/tex]
[tex]f(x) = (x-2)(x(x-3)-4(x-3))[/tex]
[tex]f(x) = (x-2)(x-3)(x-4)[/tex]
To find the roots of the function f(x) equate f(x) = 0.
we have;
[tex](x-2)(x-3)(x-4) =0[/tex]
By zero product property states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0.
x = 2 , 3 and 4.
Therefore, the roots of the given functions are; 2 , 3 and 4.
