Carlos is always playing games with his graphing calculator, but now his calculator has contracted a virus. The $\boxed{\text{TRACE}}$, $\boxed{\text{ZOOM}}$, and $\boxed{\text{WINDOW}}$ functions on his calculator are not working. He needs to solve $x^{3} + 5x^{2} – 16x – 14 = 0$, so he graphs $y = x^{3} + 5x^{2} – 16x – 14$ and sees the graph below in the standard window.

1. From the graph, what number appears to be an integer solution to the equation?

   Answer (a):

   $x = −7$

2. Verify that your answer to part (a) is a solution.

   Hint (b):

   Substitute your answer from part (a) into the original equation.

3. Since you know a solution to the equation, what is the factor associated with this solution?

   Answer (c):

   $\left(x + 7\right)$

4. Use polynomial division to determine the other factor.

   Step 1 (d):

   Divide $x^{3} + 5x^{2} − 16x − 14$ by $\left(x + 7\right)$ using an area model. 2 by 3 rectangle, labeled as follows: left edge, x, & 7, top edge left, x squared, interior top left, x cubed.

   Step 2 (d):

   Labels added to the rectangle, interior top middle, negative 2, x squared, interior bottom left, 7, x squared.

   Step 3 (d):

   Labels added to rectangle, top edge, middle, negative 2, x, interior, bottom middle, negative 14, x, interior top left, negative 2, x.

   Answer (d):

   Labels added to rectangle, top edge, right, negative 2, interior bottom right, negative 14.

   $x^{2} − 2x − 2$

   

   

   

   

5. Use your new factor to complete the following equation.

   $x^{3} + 5x^{2} – 16x – 14 = 0$
   $\left(x + 7\right)$(other factor)$= 0$

   Answer (e):

   $\left(x + 7\right)\left(x^{2} − 2x − 2\right) = 0$

6. The “other factor” leads to two other solutions to the equation. What are the two other solutions? State all three solutions to the original equation.

   Hint (f):

   Use the Quadratic Formula on the "other factor".