3x to the third power + x to the second power - 9x - 3

The expression you've given is:

\[
3x^3 + x^2 - 9x - 3
\]

There are several things we can do with this polynomial. Here's a brief breakdown of some possibilities:

### 1. **Simplification:**
   The expression is already simplified as there are no like terms to combine.

### 2. **Factoring:**
   To factor, we can attempt different methods like grouping or using the Rational Root Theorem. Let’s try factoring by grouping.

1. Group the terms:
   \[
   (3x^3 + x^2) - (9x + 3)
   \]
2. Factor out the common terms in each group:
   \[
   x^2(3x + 1) - 3(3x + 1)
   \]
3. Notice that \((3x + 1)\) is a common factor, so we factor that out:
   \[
   (3x + 1)(x^2 - 3)
   \]

Thus, the factored form of \(3x^3 + x^2 - 9x - 3\) is:
\[
(3x + 1)(x^2 - 3)
\]

### 3. **Zeros of the Polynomial:**
   To find the zeros, solve \( (3x + 1)(x^2 - 3) = 0 \).

- For \( 3x + 1 = 0 \), \( x = -\frac{1}{3} \).
- For \( x^2 - 3 = 0 \), \( x = \pm \sqrt{3} \).

So, the solutions are \( x = -\frac{1}{3}, \sqrt{3}, -\sqrt{3} \).

Would you like more details or explanations on any of these steps?

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Here are five related questions you might find useful:
1. How do you find the derivative of the polynomial \( 3x^3 + x^2 - 9x - 3 \)?
2. What are the turning points of this polynomial?
3. How can we find the integral of the expression \( 3x^3 + x^2 - 9x - 3 \)?
4. Can this polynomial have any complex roots?
5. How do you use the Rational Root Theorem to find possible rational solutions?

**Tip**: When factoring, always check for common factors or use methods like grouping and the Rational Root Theorem to simplify the process.

Discover how to factor the cubic polynomial 3x^3 + x^2 - 9x - 3 using grouping and find its zeros. This solution applies the Rational Root Theorem and quadratic formula, ideal for high school students in Grades 10-12.

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