To perform long division on the expression \(\frac{x^{10} - x^5 - 10}{x^3 - x^{12}}\), we first need to rewrite the expression in a more workable form. The denominator has the highest power term \(x^{12}\), which is larger than any power of \(x\) in the numerator. This tells us that the division will result in a fraction rather than a straightforward polynomial quotient.

However, a key observation here is that \(x^{3} - x^{12}\) can be factored as \(x^3(1 - x^9)\), which might simplify the division. Let me work through it using this approach.

### Step 1: Rewrite the expression:
\[
\frac{x^{10} - x^5 - 10}{x^3 - x^{12}} = \frac{x^{10} - x^5 - 10}{x^3(1 - x^9)}
\]

Now, divide the expression term by term:

### Step 2: Perform the long division:

#### 1st Term: 
We divide \(x^{10}\) by \(x^3(1 - x^9)\).

- Divide \(x^{10}\) by \(x^3\), resulting in \(x^7\).
- Multiply back: \(x^7 \cdot (x^3(1 - x^9)) = x^{10} - x^7\).

Now, subtract:
\[
(x^{10} - x^5 - 10) - (x^{10} - x^7) = -x^7 - x^5 - 10
\]

#### 2nd Term:
Next, divide \(-x^7\) by \(x^3(1 - x^9)\).

- Divide \(-x^7\) by \(x^3\), giving \(-x^4\).
- Multiply back: \(-x^4 \cdot (x^3(1 - x^9)) = -x^7 + x^4\).

Now, subtract:
\[
(-x^7 - x^5 - 10) - (-x^7 + x^4) = -x^5 - x^4 - 10
\]

#### 3rd Term:
Divide \(-x^5\) by \(x^3(1 - x^9)\).

- Divide \(-x^5\) by \(x^3\), giving \(-x^2\).
- Multiply back: \(-x^2 \cdot (x^3(1 - x^9)) = -x^5 + x^2\).

Now, subtract:
\[
(-x^5 - x^4 - 10) - (-x^5 + x^2) = -x^4 - x^2 - 10
\]

#### 4th Term:
Divide \(-x^4\) by \(x^3(1 - x^9)\).

- Divide \(-x^4\) by \(x^3\), giving \(-x\).
- Multiply back: \(-x \cdot (x^3(1 - x^9)) = -x^4 + x\).

Now, subtract:
\[
(-x^4 - x^2 - 10) - (-x^4 + x) = -x^2 - x - 10
\]

#### 5th Term:
Divide \(-x^2\) by \(x^3(1 - x^9)\).

- Divide \(-x^2\) by \(x^3\), resulting in a fraction, so we stop here.

### Final Answer:

The result of the long division is:
\[
x^7 - x^4 - x^2 - x + \frac{-x^2 - x - 10}{x^3(1 - x^9)}
\]

This is the quotient along with a remainder.

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Would you like further details on any part of the process?

Here are 5 related questions for deeper understanding:
1. How can we simplify higher-degree polynomials before division?
2. What is the importance of factoring the denominator in polynomial division?
3. How do we deal with remainders when performing polynomial division?
4. Can we use synthetic division for expressions with more complex denominators?
5. How does the degree of the numerator and denominator affect the quotient in polynomial division?

**Tip:** Always check if the denominator can be factored before starting long division—it can often simplify the process significantly.

x^10 - x^5 - 10 / x^3 - x^12. Use long division

Learn how to perform long division on the polynomial expression x^10 - x^5 - 10 divided by x^3 - x^12. This detailed solution includes factoring the denominator and step-by-step division. Suitable for students in Grades 11-12.

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