CPM Homework Help : A2C Problem 9-140

Carlo was trying to factor the polynomial $p\left(x\right) = x^{4} − 4x^{3} − 4x^{2} + 24x − 9$ and find all of its roots. He had already found one factor by making a guess and trying it, so he had $p\left(x\right) = \left(x − 3\right)\left(x^{3} − x^{2} − 7x + 3\right)$ . He was trying to factor $x^{3} − x^{2} − 7x + 3$ , and he had tried $\left(x + 3\right)$ , $\left(x + 1\right)$ , and $\left(x − 1\right)$ , but none worked. Then Teo came by and said, “You should look at the graph.”

How does the graph help?

Answer (a):

It shows us that $3$ is a double root which means $\left(x − 3\right)^{2}$ is a factor.

Complete the problem.

Step 1(b):

Divide $x^{3} − x^{2} − 7x + 3$ by $x − 3$ .

Step 2(b):

Use a multiplication rectangle to set it up.

Step 3(b):

Now use the quadratic formula to solve for $x$ .

Partial Answer (b):

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