Question

How do you use the rational root theorem to find the roots of `P(x) = 0.25x^2 - 12x + 23`?

Answer 1

`x=2" "` or `" "x=46`

Explanation:

First multiply by `4` to make all of the coefficients into integers:

`4P(x) = 4(0.25x^2-12x+23) = x^2-48x+92`

By the rational root theorem, any rational zeros of `x^2-48x+92` (and therefore of `P(x)`) are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `92` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2, +-4, +-23, +-46, +-92`

Trying each in turn we soon find:

`P(color(blue)(2)) = 0.25(color(blue)(2))^2-12(color(blue)(2))+23 = 1-24+23 = 0`

So `x=2` is a zero and `(x-2)` a factor:

`x^2-48x+92 = (x-2)(x-46)`

So the other zero of `P(x)` is `x=46`