Answer:

Explanation:
A cubic polynomial is given to us . And we need to find the rational roots . Rational roots means the roots which really exits and not imaginary . The given polynomial is :-

On equating it with 0 , it becomes cubic equation. That is ,
Here we will factorise out the given equation and then equate it with 0 to find the roots . Here the constant term is 6 . So the possible factors is 6 is :-

So put them one by one checking out the factors .
Put x = 2 :-

This implies that (x-2) is a factor of p(x) . Now let's divide the polynomial by (x-2) .
x-2 ) x³ + 2x² - 5x -6 ( x² +4x +3
⠀⠀⠀ (-) x³(+) -2x²
_____________
⠀⠀⠀+4x² -5x - 6
⠀⠀⠀ (-) 4x² (+) -8x
_____________
⠀⠀⠀ 3x -6
⠀⠀⠀(+) 3x(+) - 6
_______________
⠀⠀⠀⠀ 0
Hence we can write , x³ +2x² -5x -6 as ,

Equating it with 0 .
![\sf\implies x^3+2x^2-5x-6 = 0\\\\\sf\implies (x-2)(x^2 +4x+3) = 0 \\\\\sf\implies (x-2)(x^2+3x+x+3)=0 \\\\\sf\implies (x-2)[ x(x+3)+1(x+3)]=0 \\\\\sf\implies (x-2)(x+1)(x+3)=0 \\\\\sf\implies\boxed{\pink{\frak { x = 2 , (-1) , (-3) }}}](https://img.qammunity.org/2022/formulas/mathematics/college/n55fy6n6c38uup6i9ksjipkqcvuh6gx2ci.png)