Question

show that (x - 2),(x+3),(x-7) are factors of x^3 - 6x^2 + 13x + 42

btw i believe that there may be a mistake in this question. so could please cross check it. thanks!

Answer 1

Given f(x) = x^3 - 6x^2 + 13x + 42.

We know that (x - a) is a factor of f(x) if and only if f(a) = 0.

(i)

Given that (x - 2) is a factor of f(x)

f(2) = (2)^3 - 6(2)^2 + 13(2) + 42

      = 8 - 24 + 26 + 42

      = 52.


Therefore, x - 2 is not a factorof f(x).


(ii)

Given that (x + 3) is a factor of f(x).

 f(-3) = (-3)^3 - 6(-3)^2 + 13(-3) + 42

         = -27 - 54 - 39 + 42

         = -78.


Therefore (x + 3) is not a factor of (x + 3).


(iii)

Given (x - 7) is a factor of f(x).

f(7) = (7)^3 - 6(7)^2 + 13(7) + 42

      = 343 - 294 + 91 + 42
 
      = 182.


Therefore (x - 7) is not a factor of f(x).



Hope this helps!