# How do you use the rational roots theorem to find all possible zeros of x^4-5x^3-5x^2+23x+10?

Mar 19, 2016

#### Explanation:

Mar 19, 2016

$- 2 , 5 , 1 \pm \sqrt{2}$

#### Explanation:

By the rational root theorem the rational roots of the polynom can be:

$\pm 1 , \pm 2 \mathmr{and} \pm 5$

If you try the roots, you will confirm that the real roots are -2 and 5:

Now you have to divide the polynom by :

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

$\frac{{x}^{4} - 5 {x}^{3} - 5 {x}^{2} + 23 x + 10}{{x}^{2} - 3 x - 10} =$

${x}^{2} - 2 x - 1$

Use the quadratic formula to solve this part:

$x = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2 \sqrt{2}}{2}$

$x = 1 \pm \sqrt{2}$