Question

Find the integral roots of the polynomial f(x) = x³ + 6x² + 11x + 6.

Answer 1

The given polynomial is

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

Here, f(x) is a polynomial with integer coefficient and the coefficient of highest degree term is 1. So, the integer roots of f(x) are factors of 6. Which are  ±1, ±2, ±3, ±6 by observing.

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

` = -1 + 6 - 11 + 6`

` = -12 + 12`
          = 0   

Also,

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

             ` = -8 + 6 xx 4 - 22 + 6`

             ` = -8 + 42 - 22 + 6`

             `= 30 - 30`

             ` = 0` 

And similarly,

f(−3) = 0

Therefore, the integer roots of the polynomial f(x) are −1, −2, − 3.