CPM Homework Help : A2C Problem 9-121

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.

From the graph, what appears to be an integer solution to the equation?
Answer (a):
$x=−7$
Check your answer from part (a) in the equation.
Answer (b)
$y = −343 + 245 + 112 −14\\y = 0$
Since $x = −7$ is a solution to the equation, what is the factor associated with this solution?
Answer (c):
$(x+7)$
Use polynomial division to find the other factor.
Step 1(d):
Divide $x^{3} + 5x^{2} −16x −14$ by $x + 7$ using a generic rectangle.
Step 2(d):
Step 3(d):
Answer (d):
$x^{2} −2x −2$
Use your new factor to complete this equation:
$x^{3} + 5x^{2} − 16x −14 = \left(x + 7\right)(\textit{other factor}) = 0$
Answer (e):
$x ^{3} + 5x ^{2} − 16x − 14 = \left(x + 7\right)\left(x ^{2} − 2x − 2\right) = 0$
The “other factor” leads to two other solutions to the equation. Find these two new solutions and give all three solutions to the original equation.
Step (f):
Try using the quadratic formula.