Question

Using the factor theorem, what are the rational zeroes of the function `f(x) = x^4 +2x^3 - 13x^2 -38x- 24 =0`?

Answer 1

`-3;-2;-1;4`

Explanation:

We would find the rational zeroes in the factors of the known term (24), divided by the factors of the maximum degree coefficient (1):
`+-1;+-2;+-3;+-4;+-6;+-8;+-12;+-24`

Let's calculate:

f(1); f(-1);f(2);...f(-24)

we will get 0 to 4 zeroes, that's the degree of the polynomial f(x):

`f(1)=1+2-13-38-24!=0`,

then 1 isn't a zero;

`f(-1)=1-2-13+38-24=0`

then `color(red)(-1)` is a zero!

As we find a zero, we would apply the division:

`(x^4+2x^3-13x^2-38x-24)-:(x+1)`

and get remainder 0 and quotient:

`q(x)=x^3+x^2-14x-24`

and we would repeat the processing as at the beginning (with the same factors excluding 1 because it isn't a zero!)

`q(-1)=-1+1+14-24!=0`
`q(2)=8+4+28-24!=0`

`q(-2)=-8+4+28-24=0->color(red)(-2)` is a zero!
Let's divide:

`(x^3+x^2-14x-24)-:(x+2)`

and get quotient:

`x^2-x-12`

whose zeroes are `color(red)(-3)` and `color(red)(4)`