Question

How do you use the rational root theorem to find the roots of `x^4-x^3-2x^2-4x-24`?

Answer 1

Use rational root theorem to find possible rational zeros. Try the first few, then use synthetic division with the roots found to find there are no more Real ones.

`x = -2` or `x = 3`

Explanation:

Let `f(x) = x^4-x^3-2x^2-4x-24`

By the rational root theorem, any rational roots of `f(x) = 0` must be expressible in the form `p/q` in lowest terms, where `p, q in ZZ`, `q > 0`, `p` a divisor of the constant term `-24` and `q` a divisor of the coefficient `1` of the leading term.

So the only possible rational roots are:

`+-1`, `+-2`, `+-3`, `+-4`, `+-6`, `+-8`, `+-12` and `+-24`.

Let us try a few:

`f(1) = 1-1-2-4-24 = -30`
`f(-1) = 1+1-2+4-24 = -20`
`f(2) = 16-8-8-8-24 = -32`
`color(blue)(f(-2) = 16+8-8+8-24 = 0)`
`color(blue)(f(3) = 81-27-18-12-24 = 0)`

So `-2` and `3` are roots. Before going further, let's divide `f(x)` by the corresponding factors `(x+2)` and `(x-3)`

`(x+2)(x-3) = x^2-x-6`

Use synthetic division:

So `x^4-x^3-2x^2-4x-24 = (x^2-x-6)(x^2+4)`

`x^2+4` has no Real zeros since `x^2 >= 0` for all `x in RR`