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

Jan 8, 2017

$x = 2 \text{ }$ or $\text{ } x = 46$

#### Explanation:

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

$4 P \left(x\right) = 4 \left(0.25 {x}^{2} - 12 x + 23\right) = {x}^{2} - 48 x + 92$

By the rational root theorem, any rational zeros of ${x}^{2} - 48 x + 92$ (and therefore of $P \left(x\right)$) are expressible in the form $\frac{p}{q}$ 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:

$\pm 1 , \pm 2 , \pm 4 , \pm 23 , \pm 46 , \pm 92$

Trying each in turn we soon find:

$P \left(\textcolor{b l u e}{2}\right) = 0.25 {\left(\textcolor{b l u e}{2}\right)}^{2} - 12 \left(\textcolor{b l u e}{2}\right) + 23 = 1 - 24 + 23 = 0$

So $x = 2$ is a zero and $\left(x - 2\right)$ a factor:

${x}^{2} - 48 x + 92 = \left(x - 2\right) \left(x - 46\right)$

So the other zero of $P \left(x\right)$ is $x = 46$