Question

Complete the steps to solve the polynomial equation x3 – 21x = –20. According to the rational root theorem, which number is a potential root of the polynomial?

Answer 1

Zeroes : 1, 4 and -5.

Potential roots:
`\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20`.

Explanation:

The given equation is

`x^3-21x=-20`

It can be written as

`x^3+0x^2-21x+20=0`

Splitting the middle terms, we get

`x^3-x^2+x^2-x-20x+20=0`

`x^2(x-1)+x(x-1)-20(x-1)=0`

`(x-1)(x^2+x-20)=0`

Splitting the middle terms, we get

`(x-1)(x^2+5x-4x-20)=0`

`(x-1)(x(x+5)-4(x+5))=0`

`(x-1)(x+5)(x-4)=0`

Using zero product property, we get

`x-1=0\Rightarrow x=1`

`x-4=0\Rightarrow x=4`

`x+5=0\Rightarrow x=-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=\frac{\text{Factor of constant}}{\text{Factor of leading coefficient}}`

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
`\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20`.