Answer:
Zeroes : 1, 4 and -5.
Potential roots:
.
Step-by-step explanation:
The given equation is
![x^3-21x=-20](https://tex.z-dn.net/?f=x%5E3-21x%3D-20)
It can be written as
![x^3+0x^2-21x+20=0](https://tex.z-dn.net/?f=x%5E3%2B0x%5E2-21x%2B20%3D0)
Splitting the middle terms, we get
![x^3-x^2+x^2-x-20x+20=0](https://tex.z-dn.net/?f=x%5E3-x%5E2%2Bx%5E2-x-20x%2B20%3D0)
![x^2(x-1)+x(x-1)-20(x-1)=0](https://tex.z-dn.net/?f=x%5E2%28x-1%29%2Bx%28x-1%29-20%28x-1%29%3D0)
![(x-1)(x^2+x-20)=0](https://tex.z-dn.net/?f=%28x-1%29%28x%5E2%2Bx-20%29%3D0)
Splitting the middle terms, we get
![(x-1)(x^2+5x-4x-20)=0](https://tex.z-dn.net/?f=%28x-1%29%28x%5E2%2B5x-4x-20%29%3D0)
![(x-1)(x(x+5)-4(x+5))=0](https://tex.z-dn.net/?f=%28x-1%29%28x%28x%2B5%29-4%28x%2B5%29%29%3D0)
![(x-1)(x+5)(x-4)=0](https://tex.z-dn.net/?f=%28x-1%29%28x%2B5%29%28x-4%29%3D0)
Using zero product property, we get
![x-1=0\Rightarrow x=1](https://tex.z-dn.net/?f=x-1%3D0%5CRightarrow%20x%3D1)
![x-4=0\Rightarrow x=4](https://tex.z-dn.net/?f=x-4%3D0%5CRightarrow%20x%3D4)
![x+5=0\Rightarrow x=-5](https://tex.z-dn.net/?f=x%2B5%3D0%5CRightarrow%20x%3D-5)
Therefore, the zeroes of the equation are 1, 4 and -5.
According to rational root theorem, the potential root of the polynomial are
![x=\dfrac{\text{Factor of constant}}{\text{Factor of leading coefficient}}](https://tex.z-dn.net/?f=x%3D%5Cdfrac%7B%5Ctext%7BFactor%20of%20constant%7D%7D%7B%5Ctext%7BFactor%20of%20leading%20coefficient%7D%7D)
Constant = 20
Factors of constant ±1, ±2, ±4, ±5, ±10, ±20.
Leading coefficient= 1
Factors of leading coefficient ±1.
Therefore, the potential root of the polynomial are
.