Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers, fractions, and radicals, not decimal approximations.

f(x)=3x^4−x^3−6x^2

To find the real zeros of the polynomial \( f(x) = 3x^4 - x^3 - 6x^2 \), we will follow these steps:

1. **Factor the polynomial** as much as possible.
2. **Set each factor equal to zero** to find the real zeros.

### Step 1: Factor the Polynomial
First, factor out the greatest common factor (GCF), which is \( x^2 \):

\[
f(x) = x^2(3x^2 - x - 6)
\]

Now, we have \( f(x) = x^2(3x^2 - x - 6) \).

### Step 2: Set Each Factor Equal to Zero
We have two factors: \( x^2 \) and \( 3x^2 - x - 6 \).

#### Factor 1: \( x^2 = 0 \)
\[
x = 0
\]
This gives us a zero at \( x = 0 \).

#### Factor 2: \( 3x^2 - x - 6 = 0 \)

We will solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 3 \), \( b = -1 \), and \( c = -6 \). Substituting these values, we get:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-6)}}{2(3)}
\]

\[
x = \frac{1 \pm \sqrt{1 + 72}}{6}
\]

\[
x = \frac{1 \pm \sqrt{73}}{6}
\]

This gives two real zeros:

\[
x = \frac{1 + \sqrt{73}}{6}, \quad x = \frac{1 - \sqrt{73}}{6}
\]

### Conclusion
The real zeros of the polynomial \( f(x) = 3x^4 - x^3 - 6x^2 \) are:

\[
0, \quad \frac{1 + \sqrt{73}}{6}, \quad \frac{1 - \sqrt{73}}{6}
\]

If you have any questions or need more details, feel free to ask! 

Here are 5 related questions for further practice:

1. How would you find the zeros of \( f(x) = x^3 - 4x \)?
2. Can you factor \( x^4 - 16 \) into real factors?
3. What are the zeros of the polynomial \( g(x) = 2x^3 - 3x^2 - 11x + 6 \)?
4. How do you solve \( 4x^2 + 5x - 6 = 0 \) using the quadratic formula?
5. Can you find all real zeros of \( h(x) = 2x^4 - 3x^2 + 1 \)?

**Tip:** When factoring a polynomial, always start by looking for a greatest common factor (GCF).

Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers, fractions, and radicals, not decimal approximations. f(x) = 3x^4 - x^3 - 6x^2

Learn how to find all real zeros of the polynomial f(x) = 3x^4 - x^3 - 6x^2 with step-by-step instructions. This problem covers polynomial factoring and solving quadratic equations using the quadratic formula. Suitable for students in Grades 9-11.

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