Question

Show that (x − 2), (x + 3) and (x − 4) are factors of x³ − 3x² − 10x + 24.

Answer 1

Let  `f(x) = 2^3 - 3x^2 - 10x + 24` be the given polynomial.

By factor theorem,

 (x-2) , (x+3)and  (x-4) are the factor of f(x).

If   f(2) , f(-3)and f(4) are all equal to zero.

Now,

`f(2) = (2)^3 - 3(2)^2 - 10(2) + 24`

`= 8 -12 - 20 +24`

`= 32 -32`

` = 0`

also

`f(-3) = (-3)^3 -3(-3)^2 - 10(-3) + 24`

             ` = -27 -27 + 30 + 24`

             ` = -54 + 54`

             ` = 0`

And

`f(4)= (4)^3 - 3(4)^2 - 10(4) + 24`

        ` = 64 - 48 - 40 + 24`

        `= 88 - 88`

          = 0 

Hence, (x − 2), (x + 3) and (x-4) are the factor of polynomial f(x).