\#6: Factor $$ 9 x^{\wedge} 3-18 x^{\wedge} 2+x-2 $$ completely and find the'solutions. (5 Points)

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UpStudy AI · Accepted Answer

To factor the polynomial $$ 9x^3 - 18x^2 + x - 2 $$ completely and find its solutions, follow these steps:

### Step 1: Find Rational Roots
Using the **Rational Root Theorem**, possible rational roots are factors of the constant term ($$\pm1, \pm2$$) divided by factors of the leading coefficient ($$\pm1, \pm3, \pm9$$). Thus, potential rational roots are:
$$
\pm1, \pm2, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{1}{9}, \pm\frac{2}{9}
$$

### Step 2: Test Potential Roots
Let's test $$ x = 2 $$:
$$
9(2)^3 - 18(2)^2 + 2 - 2 = 72 - 72 + 2 - 2 = 0
$$
Since the result is zero, $$ x = 2 $$ is a root.

### Step 3: Factor Out $$ (x - 2) $$
Using synthetic division to divide the polynomial by $$ (x - 2) $$:

$$
\begin{array}{r|rrrr}
2 & 9 & -18 & 1 & -2 \
\hline
 & 9 & 0 & 1 & 0 \
\end{array}
$$
The quotient is $$ 9x^2 + 1 $$.

### Step 4: Factor Further
Now, the polynomial can be written as:
$$
9x^3 - 18x^2 + x - 2 = (x - 2)(9x^2 + 1)
$$
The quadratic $$ 9x^2 + 1 $$ can be factored over the complex numbers:
$$
9x^2 + 1 = (3x + i)(3x - i)
$$

### Step 5: Find All Solutions
Set each factor equal to zero:
$$
x - 2 = 0 \quad \Rightarrow \quad x = 2
$$
$$
3x + i = 0 \quad \Rightarrow \quad x = -\frac{i}{3}
$$
$$
3x - i = 0 \quad \Rightarrow \quad x = \frac{i}{3}
$$

### Final Answer
The polynomial factors completely as $$ (x - 2)(3x + i)(3x - i) $$, and its solutions are:
$$
x = 2,\quad x = \frac{i}{3},\quad \text{and} \quad x = -\frac{i}{3}
$$
The polynomial factors as $$ (x - 2)(3x + i)(3x - i) $$, and its solutions are $$ x = 2 $$, $$ x = \frac{i}{3} $$, and $$ x = -\frac{i}{3} $$.

#6: Factor completely and find the’solutions. (5 Points)

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