CPM Homework Help : CCA2 Problem 8-138

Carlo was trying to factor the polynomial $p(x)=x^4-4x^3-4x^2+24x-9$ to find all of its roots. He had already found one factor by making a guess and dividing the polynomial, so he had $p(x)=(x-3)(x^3-x^2-7x+3)$ . He was trying to factor $x^3-x^2-7x+3$ , so he had tried dividing it by $(x+3)$ , $(x+1)$ , and $(x-1)$ , but none worked without a remainder. Then Teo came by and said, “You should look at the graph.”

Continuous, curved graph, Decreasing from top left, turning at the following approximate points: low vertices: (negative 1.5, comma negative 35), & (3, comma 0), & high vertex, (1, comma 10), with x intercepts, at 3, & between negative 3 & negative 2, & between 0 & 1.

How does the graph help?
Answer (a):
It shows that $x = 3$ is a double root, so $(x - 3)$ is a repeated factor.
Complete the problem.
Step 1(b):
Divide $x^3-x^2-7x+3$ by $(x - 3)$ . 2 by 3 rectangle, labeled as follows: left edge, x, minus 3, top edge, left, x, squared, interior top, left, x squared.
Step 2(b):
Labels added: Interior bottom, left, negative 3, x, squared, interior top, middle, 2, x squared.
Step 3(b):
Labels added: top edge, middle, 2, x, interior bottom, middle, negative 6, x, interior top, right, negative x.
Step 4(b):
Labels added: top edge, right, negative 1, interior bottom, right, 3.
Step 5(b):
Now use the Quadratic Formula to solve for $x$ .
Answer (b):
$x = 3, -1 \pm \sqrt{2}$