  ### Home > CCA2 > Chapter 8 > Lesson 8.3.2 > Problem8-148

8-148.

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

1. What is $p(2)$?

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.)

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

$(x − 2)$

3. Use (b) to find another factor.

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

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

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

$(x^2 - 4x - 1)$ Labels added: top edge, right, negative 1, interior bottom, right, 2.    4. What are all the solutions of $x^3 - 6x^2 + 7x + 2 = 0$?

See part (a) for one solution.

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

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

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