https://file.aiquickdraw.com/mathbot/file/1730692166233dtyku7ty.jpg

To solve for the \( x \)- and \( y \)-intercepts of the function:

\[
f(x) = 2x^3 + 12x^2 + 16x
\]

we’ll analyze the function at key points where \( x \) and \( y \) values intercept the axes.

### Step 1: Finding the \( y \)-intercept
The \( y \)-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \):

\[
f(0) = 2(0)^3 + 12(0)^2 + 16(0) = 0
\]

So, the \( y \)-intercept is at \( (0, 0) \).

### Step 2: Finding the \( x \)-intercepts
The \( x \)-intercepts occur where \( f(x) = 0 \). Set \( f(x) = 0 \) and solve for \( x \):

\[
2x^3 + 12x^2 + 16x = 0
\]

Factor out \( 2x \):

\[
2x(x^2 + 6x + 8) = 0
\]

This gives us two factors:

1. \( 2x = 0 \) which implies \( x = 0 \).
2. \( x^2 + 6x + 8 = 0 \), a quadratic equation.

To solve \( x^2 + 6x + 8 = 0 \), factor it further:

\[
x^2 + 6x + 8 = (x + 4)(x + 2) = 0
\]

Setting each factor equal to zero:

- \( x + 4 = 0 \) gives \( x = -4 \).
- \( x + 2 = 0 \) gives \( x = -2 \).

### Summary
The intercepts are:

- \( x \)-intercepts: \( (-4, 0), (-2, 0), (0, 0) \)
- \( y \)-intercept: \( (0, 0) \)

Please enter the values:

- \( x \)-intercepts: \( -4, -2, 0 \)
- \( y \)-intercept: \( 0 \)

Do you want further details on any of the steps? 

Here are five related questions that could help deepen understanding:

1. How would you find the intercepts if the polynomial were in a different form?
2. Can a polynomial function have more than one \( y \)-intercept?
3. How would the graph look for this polynomial, and what do the intercepts tell us about it?
4. What is the multiplicity of each root for \( f(x) = 2x^3 + 12x^2 + 16x \)?
5. How do you find intercepts for more complex functions involving higher-degree terms?

**Tip:** Always start by factoring the polynomial completely to identify all roots for \( x \)-intercepts efficiently.

Find all x-intercepts and y-intercepts of the graph of the function f(x) = 2x^3 + 12x^2 + 16x.

Explore how to find the x- and y-intercepts of the polynomial function f(x) = 2x^3 + 12x^2 + 16x. This problem involves factoring and applying the Zero Product Property to determine the intercepts. 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