Question

How do you find the zeros of `f(x) = x^3 + 4x^2 - 25x - 100`?

Answer 1

There are three zeros:
`x = -5 " or " x = 5 " or " x = -4`.

Explanation:

You should try and recognize patterns in your expression that would help you factorize.

For example, here, you can notice that `25` and `100` are both dividable by `25`, so you could try to to factor `25` and find the following factorization:

`x^3 + 4x^2 - 25x - 100 = x^2(x+4) - 25(x+4) = (x^2 - 25)(x+4)`

Now you can use the identity `a^2 - b^2 = (a+b)(a-b)` to factorize further:

`... = (x+5)(x - 5)(x+4)`

Now,

`f(x) = 0`

`<=> x^3 + 4x^2 - 25x - 100 = 0`

`<=> (x+5)(x-5)(x+4) = 0`

A product is equal to zero if one or more factors is/are equal to zero:

`<=> x+5 = 0 " or " x-5 = 0 " or " x+4 = 0`

`<=> x = -5 " or " x = 5 " or " x = -4`

Answer 2

`{(x=-5),(x=5),(x=-4):}`

Explanation:

Factor by grouping:

`x^3+4x^2-25x-100`

`=(x^3+4x^2)-(25x+100)`

`=x^2(x+4)-25(x+4)`

`=(x^2-25)(x+4)`

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

This gives the solutions

`{(x=-5),(x=5),(x=-4):}`