Question

Find rational roots of the polynomial f(x) = 2x³ + x² − 7x − 6.

Answer 1

The given polynomial is

`f(x) = 2x^3 + x^2 - 7x - 6`
f(x) is a cubic polynomial with integer coefficients. If \[\frac{b}{c}\] is rational root in lowest terms, then the values of b are limited to the factors of 6 which are \[\pm 1, \pm 2, \pm 3, \pm 6\] and the values of c are limited to the factor of 2 as  \[\pm 1, \pm 2\]  Hence, the possible
rational roots are  \[\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}\].

Since, `f(2) = 2.2^3 + 2^2 - 7.2 - 6 = 0`

So, 2 is a root of the polynomial`f(x) = 2x^3 + x^2 - 7x - 6`

Now, the polynomial can be written as,

`f(x) = (x-2)(2x^2 + 5x + 3)`

Also,

`f(-1) = (-1-2) (2 - 5 + 3) = 0`

Therefore,

`f(x) = (x - 2) (x+ 1) (2x + 3)`

Hence, the rational roots of the polynomial `f(x) = 2x^3 + x^2 - 7x - 6` are 2, – 3/2 and – 1.