Find all possible rational zeros of  -3x^{3}-5x^{2}-8x+4 Write all answers as reduced fractions, and use "+-"

To find all possible rational zeros of the polynomial \( -3x^3 - 5x^2 - 8x + 4 \), we can use the **Rational Root Theorem**.

The Rational Root Theorem states that any rational root, expressed as \( \frac{p}{q} \), must have:
- \( p \), a factor of the constant term (in this case, 4).
- \( q \), a factor of the leading coefficient (in this case, -3).

### Step 1: Identify factors of the constant term and leading coefficient.

- **Constant term**: 4. The factors of 4 are \( \pm 1, \pm 2, \pm 4 \).
- **Leading coefficient**: -3. The factors of -3 are \( \pm 1, \pm 3 \).

### Step 2: Form all possible rational zeros using \( \frac{p}{q} \).

The possible rational zeros are the combinations of factors of the constant term and factors of the leading coefficient:

\[
\pm \frac{1}{1}, \pm \frac{1}{3}, \pm \frac{2}{1}, \pm \frac{2}{3}, \pm \frac{4}{1}, \pm \frac{4}{3}
\]

Thus, the possible rational zeros are:

\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}
\]

These are all the possible rational zeros of the polynomial \( -3x^3 - 5x^2 - 8x + 4 \).

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Would you like to test these values or need more details? Here are five questions to extend this concept:

1. How do you use synthetic division to test a possible rational zero?
2. What is the next step if none of the rational zeros work?
3. How does the Rational Root Theorem help simplify the process of finding roots?
4. How do irrational or complex roots relate to the Rational Root Theorem?
5. What happens if a polynomial has no rational zeros?

**Tip:** Always try substituting possible rational zeros using synthetic division to check if they are actual zeros of the polynomial.

Find all possible rational zeros of -3x^{3}-5x^{2}-8x+4. Write all answers as reduced fractions, and use '+-'

Learn how to find all possible rational zeros of the polynomial -3x^3 - 5x^2 - 8x + 4 using the Rational Root Theorem. This detailed step-by-step guide helps students understand the factors of the constant and leading coefficients, and how to use these to form possible rational zeros. Suitable for high school students in Grades 9-12.

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