Find the zeros of the polynomial f(x) = 5x

3 − 5x

2 − 10x.

To find the zeros of the polynomial \( f(x) = 5x^3 - 5x^2 - 10x \), we need to solve the equation \( f(x) = 0 \).

### Step 1: Factor out the greatest common factor (GCF).

The GCF of the terms in the polynomial is \( 5x \). So, factor out \( 5x \) from each term:

\[
f(x) = 5x(x^2 - x - 2)
\]

### Step 2: Factor the quadratic expression \( x^2 - x - 2 \).

The quadratic can be factored as:

\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Thus, the polynomial becomes:

\[
f(x) = 5x(x - 2)(x + 1)
\]

### Step 3: Set each factor equal to zero and solve for \( x \).

The equation \( 5x(x - 2)(x + 1) = 0 \) gives us three factors:

1. \( 5x = 0 \) → \( x = 0 \)
2. \( x - 2 = 0 \) → \( x = 2 \)
3. \( x + 1 = 0 \) → \( x = -1 \)

### Final Answer: The zeros of the polynomial are:
\[
x = 0, \, 2, \, -1
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you determine the GCF when factoring polynomials?
2. What is the difference between zeros and roots of a polynomial?
3. How can you confirm if a quadratic expression is factorable?
4. What are the steps to solve polynomials with higher degrees?
5. How do you solve a polynomial equation that has repeated roots?

**Tip:** Always check for common factors first when solving polynomial equations; this often simplifies the problem significantly.

Discover how to find the zeros of the polynomial 5x^3 - 5x^2 - 10x with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomials, factoring, and quadratic equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation