Question

Find all the zeros of p(x)=x^3+2x^2-3x+20

Answer 1

x = -4

Explanation:

Find the zeros of a function by setting the function equal to zero.

Basically, solve for x when p(x) = 0. Thus:

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

↓ seperate to factor out
`(x+4)`

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

↓ factor out
`(x+4)`

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

↓ simplify by combining the right part of each term

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

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

↓ split into two equations using the zero factor principle

(if
`AB = 0`, then
`A = 0` or
`B = 0`)

`x + 4 = 0`
`x^2 - 2x + 5 = 0`

`x = -4` ↓ complete the square

`x - 2x + 1 = -5 + 1`

`(x-1)^2 = -4`

`x-1 = √(-4)`

↑ no real solutions from this equation (sqrt. of negative)