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The given function is:

\[
f(x) = 3x^3 - 17x^2 + 29x - 15
\]

The problem states that there is at least one integer root, and we need to use the **Integer Root Theorem** (also known as the **Rational Root Theorem**) to find it, then proceed to find all the roots (which could be real or complex).

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem suggests that any rational root, in the form \(\frac{p}{q}\), must be a factor of the constant term (−15) divided by a factor of the leading coefficient (3).

The factors of \(-15\) (constant term) are:
\[
\pm 1, \pm 3, \pm 5, \pm 15
\]
The factors of \(3\) (leading coefficient) are:
\[
\pm 1, \pm 3
\]
Thus, the possible rational roots are:
\[
\pm 1, \pm 3, \pm 5, \pm 15, \pm \frac{1}{3}, \pm \frac{5}{3}
\]

We will test these values to find the integer root.

### Step 2: Test the possible roots
Substitute the possible integer values into \(f(x)\) to see if they result in zero.

- \(f(1) = 3(1)^3 - 17(1)^2 + 29(1) - 15 = 3 - 17 + 29 - 15 = 0\)

Thus, \(x = 1\) is a root.

### Step 3: Factor the cubic polynomial
Since \(x = 1\) is a root, we can factor \(f(x)\) as:

\[
f(x) = (x - 1)(Ax^2 + Bx + C)
\]

To find \(A\), \(B\), and \(C\), perform polynomial division of \(3x^3 - 17x^2 + 29x - 15\) by \(x - 1\), or use synthetic division.

### Step 4: Synthetic division
Perform synthetic division of \(3x^3 - 17x^2 + 29x - 15\) by \(x - 1\):

\[
\begin{array}{r|rrrr}
1 & 3 & -17 & 29 & -15 \\
  &   & 3  & -14 & 15  \\
\hline
  & 3 & -14 & 15 & 0 \\
\end{array}
\]

The quotient is \(3x^2 - 14x + 15\), so:

\[
f(x) = (x - 1)(3x^2 - 14x + 15)
\]

### Step 5: Solve the quadratic \(3x^2 - 14x + 15 = 0\)
Now solve the quadratic equation using the quadratic formula:

\[
x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(3)(15)}}{2(3)}
\]

Simplifying:

\[
x = \frac{14 \pm \sqrt{196 - 180}}{6} = \frac{14 \pm \sqrt{16}}{6} = \frac{14 \pm 4}{6}
\]

This gives two solutions:

\[
x = \frac{14 + 4}{6} = \frac{18}{6} = 3 \quad \text{and} \quad x = \frac{14 - 4}{6} = \frac{10}{6} = \frac{5}{3}
\]

### Final Roots:
The roots of \(f(x)\) are:

\[
x = 1, \quad x = 3, \quad x = \frac{5}{3}
\]

### Summary:
- The integer roots are \(x = 1\) and \(x = 3\).
- The rational root is \(x = \frac{5}{3}\).

Would you like more details on any of the steps or further clarifications?

---

Here are 5 questions related to this topic:

1. How does the Rational Root Theorem help in solving polynomial equations?
2. What is the process of synthetic division, and when is it useful?
3. Can all polynomials be factored into linear factors with real coefficients?
4. How would you solve a polynomial if no rational roots are found?
5. What is the difference between rational, irrational, and complex roots?

**Tip**: Always test the smallest factors of the constant term first when using the Rational Root Theorem to save time!

The function f(x) = 3x^3 - 17x^2 + 29x - 15 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.)

This problem explores solving the polynomial f(x) = 3x^3 - 17x^2 + 29x - 15. It involves applying the Rational Root Theorem to find an integer root, followed by factoring the polynomial using synthetic division and solving the resulting quadratic equation. Ideal for high school students in Grades 10-12.

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