Let 

*P*(*x*) = 3*x*3 − 13*x*2 − 11*x* + 5. find the zeros of P

To find the zeros of the polynomial \( P(x) = 3x^3 - 13x^2 - 11x + 5 \), we need to solve the equation \( P(x) = 0 \). This means solving the cubic equation:

\[
3x^3 - 13x^2 - 11x + 5 = 0
\]

### Steps to Find the Zeros:

1. **Rational Root Theorem**: This theorem suggests that any rational root of the polynomial equation \( P(x) = 0 \) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (5), and \( q \) is a factor of the leading coefficient (3).

   - Factors of 5 (constant term): \( \pm 1, \pm 5 \)
   - Factors of 3 (leading coefficient): \( \pm 1, \pm 3 \)

   So, the possible rational roots are: \( \pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3} \).

2. **Testing the possible rational roots**:
   We substitute these values into the polynomial to see which one (if any) makes the polynomial equal to zero.

   - For \( x = 1 \):
     \[
     P(1) = 3(1)^3 - 13(1)^2 - 11(1) + 5 = 3 - 13 - 11 + 5 = -16 \quad (\text{Not a root})
     \]
   - For \( x = -1 \):
     \[
     P(-1) = 3(-1)^3 - 13(-1)^2 - 11(-1) + 5 = -3 - 13 + 11 + 5 = 0 \quad (\text{Root found: } x = -1)
     \]

3. **Factor the polynomial**:
   Since \( x = -1 \) is a root, we can factor \( P(x) \) as:
   \[
   P(x) = (x + 1)(3x^2 - 16x + 5)
   \]

4. **Solve the quadratic equation**:
   Now, solve the quadratic equation \( 3x^2 - 16x + 5 = 0 \) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 3 \), \( b = -16 \), and \( c = 5 \).

   \[
   x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(3)(5)}}{2(3)}
   \]
   \[
   x = \frac{16 \pm \sqrt{256 - 60}}{6}
   \]
   \[
   x = \frac{16 \pm \sqrt{196}}{6}
   \]
   \[
   x = \frac{16 \pm 14}{6}
   \]

   So, the solutions are:
   \[
   x = \frac{30}{6} = 5 \quad \text{and} \quad x = \frac{2}{6} = \frac{1}{3}
   \]

### Final Answer:
The zeros of the polynomial \( P(x) = 3x^3 - 13x^2 - 11x + 5 \) are:
\[
x = -1, \quad x = \frac{1}{3}, \quad \text{and} \quad x = 5
\]

Would you like further explanations or have any other questions?

Here are some related questions you might find useful:

1. What is the Rational Root Theorem, and how does it help in solving polynomials?
2. How can synthetic division be used to factor polynomials?
3. What is the significance of the discriminant in solving quadratic equations?
4. How do you solve cubic equations that do not have rational roots?
5. What are complex roots, and how do they differ from real roots?
6. Can a polynomial have more roots than its degree? Why or why not?
7. How do you graph a polynomial function based on its roots?
8. How can you verify that a given root is correct for a polynomial equation?

**Tip:** When solving higher-degree polynomials, always check for simple rational roots first. Once a root is found, factor the polynomial and solve the remaining equation.

Learn how to find the zeros of the polynomial 3x^3 - 13x^2 - 11x + 5 with detailed step-by-step solutions. This problem involves solving cubic equations, applying the Rational Root Theorem, and factoring techniques. Suitable for high school students and advanced learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation