# What are all the possible rational zeros for f(x)=2x^3+9x^2+19x+15 and how do you find all zeros?

Jan 11, 2017

$- \frac{3}{2} , \frac{- 3 + \sqrt{11}}{2} i , \frac{- 3 - \sqrt{11}}{2} i$

#### Explanation:

x < 0 to be possible zeros for $f \left(x\right) = 2 {x}^{3} + 9 {x}^{2} + 19 x + 15$

Let we test
x=1,
$f \left(- 1\right) = 2 {\left(- 1\right)}^{3} + 9 {\left(- 1\right)}^{2} + 19 \left(- 1\right) + 15$
$f \left(- 1\right) = - 2 + 9 - 19 + 15 = 3$

x=-2,
$f \left(- 1\right) = 2 {\left(- 2\right)}^{3} + 9 {\left(- 2\right)}^{2} + 19 \left(- 2\right) + 15$
$f \left(- 1\right) = - 16 + 36 - 38 + 15 = - 3$

one of it zeros lying within -2 to -1. let we test $x = - 1.5$
$f \left(- 1.5\right) = 2 {\left(- 1.5\right)}^{3} + 9 {\left(- 1.5\right)}^{2} + 19 \left(- 1.5\right) + 15$
$f \left(- 1.5\right) = 2 \left(- 3.375\right) + 9 \left(2.25\right) + 19 \left(- 1.5\right) + 15$
$f \left(- 1.5\right) = - 6.75 + 20.25 - 28.5 + 15$
$f \left(- 1.5\right) = 0$

therefore $x = - 1.5 \mathmr{and} - \frac{3}{2}$ is one of zeros for the equation.

By using synthetic division, we get
$f \left(x\right) = 2 {x}^{3} + 9 {x}^{2} + 19 x + 15 = \left(x - \frac{3}{2}\right) \left(2 {x}^{2} + 6 x + 10\right)$

$f \left(x\right) = \left(\frac{2 x + 3}{\cancel{2}}\right) \cancel{2} \left({x}^{2} + 3 x + 5\right)$
$f \left(x\right) = \left(2 x + 3\right) \left({x}^{2} + 3 x + 5\right)$

Let consider ${x}^{2} + 3 x + 5 ,$by completing square,
${x}^{2} + 3 x + 5 = {\left(x + \frac{3}{2}\right)}^{2} - {\left(\frac{3}{2}\right)}^{2} + 5$
$= {\left(x + \frac{3}{2}\right)}^{2} - \frac{9}{4} + \frac{20}{4}$
$= {\left(x + \frac{3}{2}\right)}^{2} + \frac{11}{4}$
${\left(x + \frac{3}{2}\right)}^{2} = - \frac{11}{4}$
${\left(x + \frac{3}{2}\right)}^{2} = \frac{11}{4} {i}^{2}$
$x + \frac{3}{2} = \pm \sqrt{\left(\frac{11}{4}\right) {i}^{2}}$

$x = - \frac{3}{2} \pm \frac{\sqrt{11}}{2} i$

$x = \frac{- 3 \pm \sqrt{11}}{2} i$

$f \left(x\right) = 2 {x}^{3} + 9 {x}^{2} + 19 x + 15 = \left(2 x + 3\right) \left(x + \frac{3 + \sqrt{11}}{2} i\right) \left(x + \frac{3 - \sqrt{11}}{2} i\right)$