https://file.aiquickdraw.com/mathbot/file/1727272956667dbdrzxpc.jpg

The equation provided in the image is:

\[
x^4 - 7x^3 + 14x - 2x - 12 = 0
\]
where one of the roots is \( 1 \pm \sqrt{3} \).

I will rewrite the equation correctly, simplify it, and solve it knowing that one of the roots is \( 1 \pm \sqrt{3} \). Let's break this down and find all the roots.

### Step 1: Simplify the given equation
The equation can be simplified as:
\[
x^4 - 7x^3 + 14x^2 - 2x - 12 = 0
\]
We can now proceed to factor or use polynomial division.

### Step 2: Use the given roots \( 1 \pm \sqrt{3} \)
Since we know that \( 1 \pm \sqrt{3} \) are two of the roots, we can form a quadratic factor:
\[
(x - (1 + \sqrt{3}))(x - (1 - \sqrt{3})) = (x - 1)^2 - (\sqrt{3})^2 = x^2 - 2x + 1 - 3 = x^2 - 2x - 2
\]
This gives us a quadratic factor of \( x^2 - 2x - 2 \).

### Step 3: Perform polynomial division
Next, we divide \( x^4 - 7x^3 + 14x^2 - 2x - 12 \) by \( x^2 - 2x - 2 \) to find the other quadratic factor.

I'll compute this now.The quotient from the division is \( x^2 - 5x + 6 \), with a remainder of 0, meaning that the division is exact. Therefore, the polynomial can be factored as:

\[
(x^2 - 2x - 2)(x^2 - 5x + 6) = 0
\]

### Step 4: Solve the quadratic equations
Now, we solve both quadratics separately:

1. \( x^2 - 2x - 2 = 0 \) (this corresponds to the known roots \( 1 \pm \sqrt{3} \)).
   
2. \( x^2 - 5x + 6 = 0 \)

Let's solve the second quadratic equation \( x^2 - 5x + 6 = 0 \). This can be factored as:
\[
(x - 2)(x - 3) = 0
\]

So, the other two roots are \( x = 2 \) and \( x = 3 \).

### Final Answer
The four roots of the equation are:
\[
x = 1 + \sqrt{3}, \quad x = 1 - \sqrt{3}, \quad x = 2, \quad x = 3
\]

Would you like further details on any step or additional questions?

### Related Questions:
1. How do you perform polynomial division manually?
2. What is the method for factoring a quadratic equation?
3. Can you explain how to solve a polynomial using synthetic division?
4. What are complex roots, and how do they occur in polynomial equations?
5. How can the quadratic formula be used when factoring is not possible?

#### Tip:
For higher-degree polynomials, if some roots are known, polynomial division is a powerful tool to reduce the equation to simpler factors.

Solve x^4 - 7x^3 + 14x - 2x - 12 = 0, with one root being 1 ± √3.

Learn how to solve the polynomial equation x^4 - 7x^3 + 14x - 2x - 12 = 0 given one root as 1 ± √3. This detailed solution covers polynomial division and quadratic factoring techniques, suitable for high school students.

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