### Home > CCA2 > Chapter 8 > Lesson 8.3.2 > Problem8-138

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.”

1. How does the graph help?

It shows that $x = 3$ is a double root, so $(x - 3)$ is a repeated factor.

2. Complete the problem.

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.

Labels added: Interior bottom, left, negative 3, x, squared, interior top, middle, 2, x squared.

Labels added: top edge, middle, 2, x, interior bottom, middle, negative 6, x, interior top, right, negative x.

Labels added: top edge, right, negative 1, interior bottom, right, 3.

Now use the Quadratic Formula to solve for $x$.

$x = 3, -1 \pm \sqrt{2}$