The answer is:
(x-3)(x-1)(x+2)
Because the first term is x^3, it means that there must be 3 independent x's multiplying by each other. From there you just have to est what numbers to pair them with. If that doesn't make sense let me explain more:
If you have x^2 it will be:
(x + y) (x + z)
If you have x^3 it will be:
(x + y) (x + z) (x + g)
If you have x^4 it will be:
(x + y) (x + z) (x + g) (x + f)
etc
now all you have to do after you pick which one it matches with, it just find what the value of y, z, and g are through trial and error. I am not showing all my trial and error because it is long to type out, easier to just explain the process.
Answer: The required complete factored form of the given polynomial is [tex](x-1)(x+2)(x-3).[/tex]
Step-by-step explanation: We are given to find the complete factored form of the following cubic expression :
[tex]f(x)=x^3-2x^2-5x+6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Factor theorem : If p(a) = 0 for any polynomial p(x), then (x-a) is a factor of the polynomial p(x).
Substituting x = 1 in equation (i), we get
[tex]f(1)=1^3-2\times1^2-5\times1+6=1-2-5+6=0.[/tex]
So, (x-1) is a factor of f(x).
Therefore, we have
[tex]f(x)\\\\=x^3-2x^2-5x+6\\\\=x^2(x-1)-x(x-1)-6(x-1)\\\\=(x-1)(x^2-x-6)\\\\=(x-1)(x^2-3x+2x-6)\\\\=(x-2)(x(x-3)+2(x-3))\\\\=(x-1)(x+2)(x-3).[/tex]
Thus, the required complete factored form of the given polynomial is [tex](x-1)(x+2)(x-3).[/tex]
