Question

f(x) = x³ − 6x² + 11x − 6, g(x) = x² − 3x + 2

Answer 1

It is given that f(x) = x³ − 6x² + 11x − 6, and g(x) = x² − 3x + 2

We have

g(x) = x² − 3x + 2

     ` = x^2 - 2x + x + 2`

     ` = (x - 2) (x-1)`

\[\Rightarrow \left( x - 2 \right)\]

and (x − 1) are factor of g(x) by the factor theorem.

To prove that (x − 2) and (x − 1) are the factor of f(x).

It is sufficient to show that f(2) and f(1) both are equal to zero.

`f(2) = (2)^3 - 6(2)^3 + 11(2) - 6`

       ` = 8 - 23 + 22 - 6`

       ` = 30 - 30`

 f(2) = 0

And

`f(1) = (1)^3 - 6(1)^2 + 11(1)- 6`

` = 1-6 + 11 - 6`

`  = 12 - 12`

 f (1) = 0 

Hence, g(x) is the factor of the polynomial f(x).