Math Problem Statement

Find all zeros of the polynomial P(x) = x^3 + 3x^2 - x - 3. The zeros are x1, x2, and x3, with x1 ≤ x2 ≤ x3.

Solution

To solve the problem of finding all zeros of the polynomial P(x)=x3+3x2−x−3P(x) = x^3 + 3x^2 - x - 3, we follow these steps:

Step 1: Rational Root Theorem

The possible rational roots of the polynomial are the factors of the constant term (-3) divided by the factors of the leading coefficient (1). Thus, the possible roots are: ±1,±3\pm 1, \pm 3

Step 2: Test Possible Roots

We substitute these possible roots into P(x)P(x) to see which one is a root.

  • Test x=1x = 1: P(1)=13+3(12)−1−3=1+3−1−3=0P(1) = 1^3 + 3(1^2) - 1 - 3 = 1 + 3 - 1 - 3 = 0 Thus, x=1x = 1 is a root.

Step 3: Synthetic Division

Divide P(x)P(x) by (x−1)(x - 1) using synthetic division:

1 & 1 & 3 & -1 & -3 \\ & & 1 & 4 & 3 \\ \hline & 1 & 4 & 3 & 0 \\ \end{array}$$ The quotient is $$x^2 + 4x + 3$$, so: $$P(x) = (x - 1)(x^2 + 4x + 3)$$ ### Step 4: Factor the Quadratic Factor $$x^2 + 4x + 3$$: $$x^2 + 4x + 3 = (x + 3)(x + 1)$$ ### Step 5: Write All Zeros The complete factorization is: $$P(x) = (x - 1)(x + 3)(x + 1)$$ Thus, the zeros are: $$x_1 = -3, \quad x_2 = -1, \quad x_3 = 1$$ ### Final Answer: The zeros of the polynomial are: $$x_1 = -3, \quad x_2 = -1, \quad x_3 = 1$$ Let me know if you'd like a detailed explanation of any step or have more questions! Here's a list of related questions: 1. What is the Rational Root Theorem, and how is it applied? 2. How does synthetic division work for polynomial division? 3. Why does factoring a quadratic give its roots? 4. Can this process be applied to higher-degree polynomials? 5. How can we confirm all roots satisfy the original polynomial? **Tip:** Always verify all roots by substituting them back into the polynomial to ensure accuracy.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Polynomial Zeros
Factoring Polynomials
Synthetic Division

Formulas

Rational Root Theorem
Factoring quadratic expressions

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 9-12

Related Recommendation

Finding Zeros of the Polynomial P(x) = x^3 - x - 6

Finding Zeros of the Polynomial P(x) = x^3 + 2x^2 - x - 2

Solving the Polynomial Equation x^3 + 3x^2 - 4 Using Rational Root Theorem

Finding Real Zeros of a Polynomial x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3

Find the Real Zeros of the Polynomial x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3