# How do you use the rational root theorem to find the roots of 3x^4+2x^3-9x^2-12x-4=0?

Dec 7, 2015

Use the rational root theorem to identify the possible roots, try some and narrow down the search to find roots $- \frac{2}{3}$, $- 1$, $- 1$ and $2$.

#### Explanation:

$f \left(x\right) = 3 {x}^{4} + 2 {x}^{3} - 9 {x}^{2} - 12 x - 4$

By the rational root theorem any rational roots of $f \left(x\right) = 0$ are expressible in lowest terms as $\frac{p}{q}$ for some integers $p$ and $q$, where $p$ is a divisor of the constant term $4$ and $q$ is a divisor of the coefficient $3$ of the leading term.

So the possible rational roots are:

$\pm \frac{1}{3}$, $\pm \frac{2}{3}$, $\pm 1$, $\pm \frac{4}{3}$, $\pm 2$, $\pm 4$

Try:

$f \left(\frac{1}{3}\right) = \frac{1}{27} + \frac{2}{27} - 1 - 4 - 4 = \frac{1}{9} - 9 = - \frac{80}{9}$

$f \left(- \frac{1}{3}\right) = \frac{1}{27} - \frac{2}{27} - 1 + 4 - 4 = - \frac{28}{27}$

$f \left(\frac{2}{3}\right) = \frac{16}{27} + \frac{16}{27} - 4 - 8 - 4 = \frac{32}{27}$

$f \left(- \frac{2}{3}\right) = \frac{16}{27} - \frac{16}{27} - 4 + 8 - 4 = 0$

So $x = - \frac{2}{3}$ is a root. This 'uses up' one of the possible factors of $2$ for $p$ and the factor of $3$ for $q$. So the only other possible rational roots are:

$\pm 1$, $\pm 2$

Try:

$f \left(1\right) = 3 + 2 - 9 - 12 - 4 = - 20$

$f \left(- 1\right) = 3 - 2 - 9 + 12 - 4 = 0$

$f \left(2\right) = 48 + 16 - 36 - 24 - 4 = 0$

$f \left(- 2\right) = 48 - 16 - 36 + 24 - 4 = 16$

So we have identified $3$ rational roots out of $4$ roots, viz $- \frac{2}{3}$, $- 1$ and $2$. The fourth root must also be rational and having 'used up' the factor of $2$ for $p$, it must also be $- 1$.

To check, we can multiply out the corresponding factors:

$\left(3 x + 2\right) \left(x + 1\right) \left(x + 1\right) \left(x - 2\right)$

$= \left(3 {x}^{2} + 5 x + 2\right) \left({x}^{2} - x - 2\right)$

$= 3 {x}^{4} + \left(5 - 3\right) {x}^{3} - \left(6 + 5 - 2\right) {x}^{2} - \left(10 + 2\right) x - 4$

$= 3 {x}^{4} + 2 {x}^{3} - 9 {x}^{2} - 12 x - 4$