Find the zeros of the function algebraically. (Enter your answers as a comma-separated list.)

*f*(*x*) = *x*3 − 5*x*2 − 16*x* + 80

To find the zeros of the function \( f(x) = x^3 - 5x^2 - 16x + 80 \), we need to solve the equation:

\[
f(x) = 0
\]

### Step 1: Use the Rational Root Theorem
According to the Rational Root Theorem, possible rational roots are factors of the constant term (80) divided by factors of the leading coefficient (1). So, possible rational roots are:

\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 16, \pm 20, \pm 40, \pm 80
\]

We can test these values by substitution to see if any satisfy \( f(x) = 0 \).

### Step 2: Trial and Error for Rational Roots
Let's test \( x = 4 \):

\[
f(4) = 4^3 - 5(4^2) - 16(4) + 80 = 64 - 80 - 64 + 80 = 0
\]

Thus, \( x = 4 \) is a root.

### Step 3: Factor the polynomial
Since \( x = 4 \) is a root, we can divide \( f(x) \) by \( (x - 4) \). We use synthetic division to do this.

#### Synthetic division setup:
\[
\begin{array}{r|rrrr}
4 & 1 & -5 & -16 & 80 \\
  &   & 4  & -4  & -80 \\
\hline
  & 1 & -1 & -20 & 0 \\
\end{array}
\]

The quotient is \( x^2 - x - 20 \).

### Step 4: Solve the quadratic equation
Now, solve \( x^2 - x - 20 = 0 \) using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 80}}{2}
\]
\[
x = \frac{1 \pm \sqrt{81}}{2}
\]
\[
x = \frac{1 \pm 9}{2}
\]

Thus, the solutions are:

\[
x = \frac{1 + 9}{2} = 5, \quad x = \frac{1 - 9}{2} = -4
\]

### Step 5: Final answer
The zeros of the function are:

\[
x = 4, \, x = 5, \, x = -4
\]

The solution is \( x = 4, 5, -4 \).

Would you like further details or clarifications?

Here are some related questions to explore:
1. How do you verify the zeros of a polynomial?
2. What is the Rational Root Theorem and how is it applied?
3. Can you always solve cubic functions algebraically?
4. How does synthetic division work?
5. What is the quadratic formula and when is it used?

**Tip:** Synthetic division is a useful tool for dividing polynomials, especially when factoring cubic or higher-degree functions.

Find the zeros of the function algebraically. (Enter your answers as a comma-separated list.) f(x) = x^3 − 5x^2 − 16x + 80

Learn how to find the zeros of the cubic function f(x) = x^3 − 5x^2 − 16x + 80 using algebraic methods. This step-by-step solution involves the Rational Root Theorem, synthetic division, and solving a quadratic equation. Perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation