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To solve each part of question 9, we need to use polynomial long division or reverse the division process by finding the divisor, given the dividend, quotient, and remainder.

The relationship is based on:

\[
\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}
\]

We can rearrange this to solve for the divisor:

\[
\text{Divisor} = \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}
\]

Let's go through each part.

---

### (a)
- **Dividend:** \( 5x^3 + x^2 + 3 \)
- **Quotient:** \( 5x^2 - 14x + 42 \)
- **Remainder:** \( -123 \)

Substitute the values into the formula:

\[
5x^3 + x^2 + 3 = (\text{Divisor} \times (5x^2 - 14x + 42)) - 123
\]

Solve for the divisor by dividing:

\[
\text{Divisor} = \frac{(5x^3 + x^2 + 3 + 123)}{(5x^2 - 14x + 42)}
\]
\[
\text{Divisor} = \frac{(5x^3 + x^2 + 126)}{(5x^2 - 14x + 42)}
\]

Perform polynomial division to find the divisor.

---

### (b)
- **Dividend:** \( 10x^4 - x^2 + 20x - 2 \)
- **Quotient:** \( 10x^3 - 100x^2 + 999x - 9970 \)
- **Remainder:** \( 99,698 \)

Substitute into the relationship:

\[
10x^4 - x^2 + 20x - 2 = (\text{Divisor} \times (10x^3 - 100x^2 + 999x - 9970)) + 99,698
\]

Solve for the divisor:

\[
\text{Divisor} = \frac{(10x^4 - x^2 + 20x - 2 - 99,698)}{(10x^3 - 100x^2 + 999x - 9970)}
\]

Simplify and perform polynomial division to find the divisor.

---

### (c)
- **Dividend:** \( x^4 + x^3 - 10x^2 + 2x - 8 \)
- **Quotient:** \( x^3 - 3x^2 + 2x - 8 \)
- **Remainder:** 31

\[
x^4 + x^3 - 10x^2 + 2x - 8 = (\text{Divisor} \times (x^3 - 3x^2 + 2x - 8)) + 31
\]

Solve for the divisor:

\[
\text{Divisor} = \frac{(x^4 + x^3 - 10x^2 + 2x - 8 - 31)}{(x^3 - 3x^2 + 2x - 8)}
\]

Perform polynomial division to find the divisor.

---

### (d)
- **Dividend:** \( x^3 + x^2 + 7x - 7 \)
- **Quotient:** \( x^2 + 3x + 13 \)
- **Remainder:** 19

\[
x^3 + x^2 + 7x - 7 = (\text{Divisor} \times (x^2 + 3x + 13)) + 19
\]

Solve for the divisor:

\[
\text{Divisor} = \frac{(x^3 + x^2 + 7x - 7 - 19)}{(x^2 + 3x + 13)}
\]

Simplify and perform polynomial division to find the divisor.

---

Would you like a detailed step-by-step breakdown for any specific part of this solution?

Here are five related questions for further exploration:
1. How do you perform polynomial long division step-by-step?
2. What are the key differences between polynomial division and synthetic division?
3. How can you verify the result of a polynomial division problem?
4. What is the remainder theorem and how does it apply to these problems?
5. How can you factor polynomials efficiently when solving division problems?

**Tip:** In polynomial division, always ensure the degrees of both the dividend and divisor are in descending order to avoid confusion.

Each dividend was divided by another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the divisor).

Solve polynomial division problems where you're given the dividend, quotient, and remainder. Learn how to use the formula Dividend = (Divisor × Quotient) + Remainder to find the divisor. This problem is suitable for students in Grades 10-12.

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