Question

Using factor theorem, factorize each of the following polynomials:
x³ + 6x² + 11x + 6

Answer 1

Let f(x) = x³ + 6x² + 11x + 6 be the given polynomial.

Now, put the x =-1we get

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

             ` = -1 + 6 -11 + 6`

             ` = -12 + 12`

             ` = 0`

Therefore, (x +1)is a factor of f(x).

Now,

`f(x) = x^2 (x+1) + 5x (x +1) + 6 (x+1)`

        ` = (x +1){x^2 + 5x + 6}`

        ` = (x+1) {x^2 + 3x + 2x + 6}`

        ` = (x +1) (x+2)(x+3)` 

`f(x) = x^2 (x+1) + 5`

Hence, (x+1)(x+2)(x+3)are the factors of f(x).