# How do you use the Intermediate Value Theorem to show that the polynomial function  2x^3 + x^2 +2 has a root in the interval [-2, -1]?

##### 1 Answer
Oct 19, 2015

See the explanation.

#### Explanation:

$f \left(x\right) = 2 {x}^{3} + {x}^{2} + 2$ is continuous function over $R$ and hence, over $\left[- 2 , - 1\right]$.

$f \left(- 2\right) = 2 {\left(- 2\right)}^{3} + {\left(- 2\right)}^{2} + 2 = - 16 + 4 + 2 = - 10$
$f \left(- 1\right) = 2 {\left(- 1\right)}^{3} + {\left(- 1\right)}^{2} + 2 = - 2 + 1 + 2 = 1$

So, $f \left(x\right)$ has a root on $\left[- 2 , - 1\right]$.