# How do you determine p(c) given p(x)=x^4+x^3-6x^2-7x-7 and c=-sqrt7?

Jul 6, 2017

$- 14$

#### Explanation:

Grab your calculator... Because this is a REAL doozy....

Since:
$c = - \sqrt{7}$
And:
$p \left(x\right) = {x}^{4} + {x}^{3} - 6 {x}^{2} - 7 x - 7$

Substitute
$c = - \sqrt{7}$
Into $p \left(x\right)$, which is:
p(x)= (-sqrt7)^4+(-sqrt7)^3-6(-sqrt7)^2 - 7(-sqrt7)-7

Simplifying gives

${\left(- 7\right)}^{2} - {\left(7\right)}^{\frac{3}{2}} - 6 \left(7\right) - 7 {\left(7\right)}^{\frac{1}{2}} - 7$

Using a calculator,
${7}^{\frac{3}{2}} = 18.520$

And
${7}^{\frac{1}{2}} = 2.646$

Solving all the stuffs
$49 - 18.520 - 42 + 7 \left(2.646\right) - 7$

$49 - 42 - 7$

$= 0$

Yay! DONE!

Jul 6, 2017

$0$

#### Explanation:

What we can do is plug in the value $- \sqrt{7}$ in for each $x$:

$p \left(c\right) = {\left(- \sqrt{7}\right)}^{4} + {\left(- \sqrt{7}\right)}^{3} - 6 {\left(- \sqrt{7}\right)}^{2} - 7 \left(- \sqrt{7}\right) - 7$

$= 49 - 7 \sqrt{7} - 42 + 7 \sqrt{7} - 7$

= 49 - 42 - 7 = color(blue)(0

Thus, the value "$- \sqrt{7}$" is a zero of this polynomial function.