Question The function $$ f(x)=3 x^{3}-2 x^{2}-3 x+2 $$ has at least one integer root. Use the integer root theorem to find that root, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.) Answer Attempt 1 out of 10

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To find the integer roots of the polynomial function $$ f(x) = 3x^3 - 2x^2 - 3x + 2 $$ using the Integer Root Theorem, we will follow these steps:

1. **Identify the possible integer roots**: According to the Integer Root Theorem, the possible rational roots of a polynomial are given by the factors of the constant term divided by the factors of the leading coefficient.

- The constant term is $$ 2 $$, and its factors are $$ \pm 1, \pm 2 $$.
   - The leading coefficient is $$ 3 $$, and its factors are $$ \pm 1, \pm 3 $$.

Therefore, the possible integer roots are:
   $$
   \pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}
   $$

2. **Test the possible integer roots**: We will evaluate $$ f(x) $$ at each of the integer candidates $$ \pm 1 $$ and $$ \pm 2 $$.

Let's calculate $$ f(1) $$, $$ f(-1) $$, $$ f(2) $$, and $$ f(-2) $$:

- For $$ f(1) $$:
$$
f(1) = 3(1)^3 - 2(1)^2 - 3(1) + 2
$$

- For $$ f(-1) $$:
$$
f(-1) = 3(-1)^3 - 2(-1)^2 - 3(-1) + 2
$$

- For $$ f(2) $$:
$$
f(2) = 3(2)^3 - 2(2)^2 - 3(2) + 2
$$

- For $$ f(-2) $$:
$$
f(-2) = 3(-2)^3 - 2(-2)^2 - 3(-2) + 2
$$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$3\left(-1\right)^{3}-2\left(-1\right)^{2}-3\left(-1\right)+2$$
- step1: Evaluate the power:
  $$3\left(-1\right)^{3}-2\times 1-3\left(-1\right)+2$$
- step2: Multiply the terms:
  $$-3-2\times 1-3\left(-1\right)+2$$
- step3: Multiply:
  $$-3-2-3\left(-1\right)+2$$
- step4: Simplify:
  $$-3-2+3+2$$
- step5: Simplify:
  $$-2+2$$
- step6: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ 3(-2)^3 - 2(-2)^2 - 3(-2) + 2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3\left(-2\right)^{3}-2\left(-2\right)^{2}-3\left(-2\right)+2$$
- step1: Multiply the terms:
  $$-24-2\left(-2\right)^{2}-3\left(-2\right)+2$$
- step2: Multiply the terms:
  $$-24-2^{3}-3\left(-2\right)+2$$
- step3: Multiply the numbers:
  $$-24-2^{3}+6+2$$
- step4: Evaluate the power:
  $$-24-8+6+2$$
- step5: Calculate:
  $$-24$$
Calculate or simplify the expression $$ 3(1)^3 - 2(1)^2 - 3(1) + 2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3\times 1^{3}-2\times 1^{2}-3\times 1+2$$
- step1: Evaluate the power:
  $$3\times 1-2\times 1^{2}-3\times 1+2$$
- step2: Evaluate the power:
  $$3\times 1-2\times 1-3\times 1+2$$
- step3: Multiply:
  $$3-2\times 1-3\times 1+2$$
- step4: Multiply:
  $$3-2-3\times 1+2$$
- step5: Multiply:
  $$3-2-3+2$$
- step6: Simplify:
  $$-2+2$$
- step7: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ 3(2)^3 - 2(2)^2 - 3(2) + 2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3\times 2^{3}-2\times 2^{2}-3\times 2+2$$
- step1: Multiply the terms:
  $$24-2\times 2^{2}-3\times 2+2$$
- step2: Calculate:
  $$24-2^{3}-3\times 2+2$$
- step3: Multiply the numbers:
  $$24-2^{3}-6+2$$
- step4: Evaluate the power:
  $$24-8-6+2$$
- step5: Calculate:
  $$12$$
The evaluations of the function $$ f(x) $$ at the integer candidates are as follows:

- $$ f(1) = 0 $$
- $$ f(-1) = 0 $$
- $$ f(2) = 12 $$
- $$ f(-2) = -24 $$

Since both $$ f(1) $$ and $$ f(-1) $$ equal $$ 0 $$, we have found two integer roots: $$ x = 1 $$ and $$ x = -1 $$.

Next, we will factor the polynomial $$ f(x) $$ using one of the integer roots. Let's use $$ x = 1 $$ to perform synthetic division on $$ f(x) $$.

### Synthetic Division with $$ x = 1 $$

The coefficients of $$ f(x) $$ are $$ [3, -2, -3, 2] $$.

Performing synthetic division:

$$
\begin{array}{r|rrrr}
1 & 3 & -2 & -3 & 2 \
  &   & 3 & 1 & -2 \
\hline
  & 3 & 1 & -2 & 0 \
\end{array}
$$

The result of the synthetic division is $$ 3x^2 + x - 2 $$.

Now, we will factor $$ 3x^2 + x - 2 $$ to find the remaining roots.

### Factoring $$ 3x^2 + x - 2 $$

To factor $$ 3x^2 + x - 2 $$, we can look for two numbers that multiply to $$ 3 \times (-2) = -6 $$ and add to $$ 1 $$. The numbers $$ 3 $$ and $$ -2 $$ work.

We can rewrite the middle term:
$$
3x^2 + 3x - 2x - 2
$$

Now, we can factor by grouping:
$$
(3x^2 + 3x) + (-2x - 2) = 3x(x + 1) - 2(x + 1) = (3x - 2)(x + 1)
$$

Thus, we have:
$$
f(x) = (x - 1)(3x - 2)(x + 1)
$$

### Finding the Roots

Now we can find the roots from the factors:
1. From $$ x - 1 = 0 $$, we get $$ x = 1 $$.
2. From $$ 3x - 2 = 0 $$, we get $$ x = \frac{2}{3} $$.
3. From $$ x + 1 = 0 $$, we get $$ x = -1 $$.

### Summary of Roots

The roots of the polynomial $$ f(x) = 3x^3 - 2x^2 - 3x + 2 $$ are:
- Integer roots: $$ 1 $$ and $$ -1 $$
- Rational root: $$ \frac{2}{3} $$

Since all roots found are real, there are no complex roots in this case.

Thus, the complete set of roots is:
- $$ 1, -1, \frac{2}{3} $$
The integer roots are $$ x = 1 $$ and $$ x = -1 $$. The rational root is $$ x = \frac{2}{3} $$. There are no complex roots.

Question
The function has at least one integer root.
Use the integer root theorem to find that root, then proceed to find all
complex roots. (Note: roots may be integer, rational, irrational, and/or
complex.)
Answer Attempt 1 out of 10

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Question The function has at least one integer root. Use the integer root theorem to find that root, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.) Answer Attempt 1 out of 10