Given the polynomial $p(x) = x^3 - 6x^2 + 7x + 2$.

1. What is $p(2)$?

   Hint (a):

   Substitute $2$ into the equation for every $x$.

2. Use the Remainder Theorem to find one factor of $x^3 − 6x^2 + 7x + 2$. (See the Math Notes box in Lesson 8.3.2 above.)

   Hint (b):

   Since $p(2) = 0$, $x = 2$ is the zero of the function. What is the corresponding factor?

   Answer (b):

   $(x − 2)$

3. Use (b) to find another factor.

   Step 1(c):

   Try using a generic rectangle. A 2 by 3 rectangle, labeled as follows: left edge, x, minus 2, interior top, left, x, cubed.

   Step 2(c):

   Labels added: top edge, left, x, squared, interior bottom, left, negative 2, x, squared, interior top, middle, negative 4, x squared.

   Step 3(c):

   Labels added: top edge, middle, negative 4, x, interior bottom, middle, 8, x, interior top, right, negative x.

   Answer (c):

   $(x^2 - 4x - 1)$ Labels added: top edge, right, negative 1, interior bottom, right, 2.

   

   

   

   

4. What are all the solutions of $x^3 - 6x^2 + 7x + 2 = 0$?

   Hint (d):

   See part (a) for one solution.

   Step (d):

   Use the Quadratic Formula to find the solutions to $0 = x^2 - 4x - 1$.

   $x=\frac{4 \pm \sqrt{16 + 4}}{2}$

   Answer (d):

   $x = 2, 2 \pm \sqrt{5}$