https://file.aiquickdraw.com/mathbot/file/1726661015375kw7k0u3s.jpg

To explain the method shown in the image, I can recreate a similar example for you.

Let's break down what is happening. The example shows how to solve a quartic polynomial equation using **synthetic division** and factoring. Here's a step-by-step recreation:

### Problem:
Solve the quartic equation:

\[
x^4 - 2x^3 - 7x^2 + 8x + 12 = 0
\]

### Steps:

#### 1. Find Possible Roots (Rational Root Theorem):
The possible rational roots are factors of the constant term (12) divided by factors of the leading coefficient (1). These are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]

#### 2. Use Synthetic Division:
We begin by testing one of the possible roots using synthetic division.

##### First, try \( x = -1 \):
\[
\begin{array}{r|rrrrr}
-1 & 1 & -2 & -7 & 8 & 12 \\
   &   & -1 & 3 & 4 & -12 \\
\hline
   & 1 & -3 & -4 & 12 & 0 \\
\end{array}
\]
Since the remainder is 0, \( x = -1 \) is a root, and the quotient is \( x^3 - 3x^2 - 4x + 12 \).

#### 3. Repeat Synthetic Division with Another Root:
Now, take the quotient \( x^3 - 3x^2 - 4x + 12 \), and try another possible root.

##### Try \( x = -2 \):
\[
\begin{array}{r|rrrr}
-2 & 1 & -3 & -4 & 12 \\
   &   & -2 & 10 & -12 \\
\hline
   & 1 & -5 & 6 & 0 \\
\end{array}
\]
Again, the remainder is 0, so \( x = -2 \) is a root, and the quotient is \( x^2 - 5x + 6 \).

#### 4. Solve the Remaining Quadratic:
Now, solve the quadratic \( x^2 - 5x + 6 = 0 \) by factoring:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Thus, the remaining roots are \( x = 2 \) and \( x = 3 \).

### Final Solution:
The roots of the equation are:
\[
x = -1, -2, 2, 3
\]

Would you like more details on any part of this process or have any specific questions?

Here are 5 related questions:
1. What is the Rational Root Theorem, and how is it applied in synthetic division?
2. How does synthetic division simplify solving higher-degree polynomials?
3. Why is it important to factor the quadratic at the end?
4. How would you solve the equation if no rational roots were found?
5. Can synthetic division be applied to polynomials with non-integer coefficients?

### Tip:
Synthetic division is a powerful tool for simplifying polynomials. Always start by testing simple roots like \( \pm 1 \), as they can quickly simplify complex equations!

Solve the quartic equation x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 using synthetic division.

Learn how to solve the quartic equation x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 using synthetic division. This step-by-step guide covers key algebraic concepts, such as polynomial division and factoring. Suitable for students in Grades 9-12.

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