# Is -1/2 a zero of the polynomial f(x)=4x^4+9x^2+9x+2?

Nov 1, 2016

Yes

#### Explanation:

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

So

$f \left(\textcolor{b l u e}{- \frac{1}{2}}\right) = \left(\left(4 {\left(\textcolor{b l u e}{- \frac{1}{2}}\right)}^{2} + 9\right) \left(\textcolor{b l u e}{- \frac{1}{2}}\right) + 9\right) \left(\textcolor{b l u e}{- \frac{1}{2}}\right) + 2$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- \frac{1}{2}}\right)} = \left(\left(1 + 9\right) \left(\textcolor{b l u e}{- \frac{1}{2}}\right) + 9\right) \left(\textcolor{b l u e}{- \frac{1}{2}}\right) + 2$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- \frac{1}{2}}\right)} = \left(- 5 + 9\right) \left(\textcolor{b l u e}{- \frac{1}{2}}\right) + 2$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- \frac{1}{2}}\right)} = - 2 + 2$

$\textcolor{w h i t e}{f \left(\textcolor{w h i t e}{- \frac{1}{2}}\right)} = 0$

Yes, $f \left(- \frac{1}{2}\right) = 0$