Solve this for me in x and y intercept show me the full solution. g(x)= 6x⁴-2x³-6x²+2x in the table of values of -2, -1, 0, 1

Let's solve the given function \( g(x) = 6x^4 - 2x^3 - 6x^2 + 2x \) and find both the **x-intercepts** and **y-intercepts**. Afterward, we'll evaluate the function at \( x = -2, -1, 0, 1 \) to fill the table of values.

### 1. **Finding the y-intercept**

The y-intercept occurs when \( x = 0 \).

\[
g(0) = 6(0)^4 - 2(0)^3 - 6(0)^2 + 2(0) = 0
\]

Thus, the **y-intercept** is \( (0, 0) \).

### 2. **Finding the x-intercepts**

The x-intercepts occur when \( g(x) = 0 \). This leads to the equation:

\[
6x^4 - 2x^3 - 6x^2 + 2x = 0
\]

Factor the equation:

\[
2x(3x^3 - x^2 - 3x + 1) = 0
\]

Now, set each factor equal to zero:
1. \( 2x = 0 \)
2. \( 3x^3 - x^2 - 3x + 1 = 0 \)

From the first factor, \( 2x = 0 \), we find \( x = 0 \). This gives us the x-intercept \( (0, 0) \), which we already found as the y-intercept.

Now, solve the cubic equation \( 3x^3 - x^2 - 3x + 1 = 0 \). This equation requires more advanced techniques to find roots, but for now, let's focus on solving this graphically.

### 3. **Table of Values**

Next, let's evaluate the function \( g(x) = 6x^4 - 2x^3 - 6x^2 + 2x \) at \( x = -2, -1, 0, 1 \).

\[
g(-2) = 6(-2)^4 - 2(-2)^3 - 6(-2)^2 + 2(-2)
\]
\[
= 6(16) - 2(-8) - 6(4) + 2(-2) = 96 + 16 - 24 - 4 = 84
\]

\[
g(-1) = 6(-1)^4 - 2(-1)^3 - 6(-1)^2 + 2(-1)
\]
\[
= 6(1) - 2(-1) - 6(1) + 2(-1) = 6 + 2 - 6 - 2 = 0
\]

\[
g(0) = 6(0)^4 - 2(0)^3 - 6(0)^2 + 2(0) = 0
\]

\[
g(1) = 6(1)^4 - 2(1)^3 - 6(1)^2 + 2(1)
\]
\[
= 6(1) - 2(1) - 6(1) + 2(1) = 6 - 2 - 6 + 2 = 0
\]

### 4. **Summary of Table of Values**

| \( x \)  | \( g(x) \)  |
|-------|----------|
| -2    | 84       |
| -1    | 0        |
|  0    | 0        |
|  1    | 0        |

### 5. **X-intercepts**

From the table and factoring, we can see that \( g(x) = 0 \) at \( x = -1, 0, 1 \), so the x-intercepts are \( (-1, 0), (0, 0), (1, 0) \).

Would you like further details or clarification on any part of the solution?

Here are 5 related questions:
1. How do we fully factor a cubic equation like \( 3x^3 - x^2 - 3x + 1 = 0 \)?
2. Can a function have the same point as both an x-intercept and y-intercept?
3. What is the significance of higher-degree polynomials in graphing?
4. How do the coefficients affect the shape of a polynomial graph?
5. What is the process for finding turning points in a fourth-degree polynomial?

**Tip**: For solving cubic equations, you can use methods like synthetic division or the Rational Root Theorem to factor them into simpler components.

Discover how to solve the polynomial function g(x) = 6x⁴ - 2x³ - 6x² + 2x for both x and y intercepts and fill in a table of values for x = -2, -1, 0, 1. This solution involves factoring polynomials and finding intercepts, perfect for Grades 10-12 math learners.

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