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The exercises in the image focus on factoring polynomials, the remainder theorem, and using polynomial division. Here's how we can solve them:

### 1. **Find the factors of \( 4x^3 - 10x^2 - 3x + 9 \) using the remainder and factor theorem.**
   - **Factor Theorem:** If \( f(c) = 0 \), then \( (x - c) \) is a factor of \( f(x) \).
   - Use synthetic division or the remainder theorem to identify potential factors.

### 2. **What are the factors of \( x^3 - 3x^2 - 5x + 15 \)?**
   - Similarly, apply the remainder theorem and synthetic division.
   - Look for factors like \( (x - c) \) where \( f(c) = 0 \).

### 3. **Find the remainder when \( P(x) = x^4 - 3x^2 - 4x + 16 \) is divided by \( x - 4 \).**
   - Using the **Remainder Theorem**, substitute \( x = 4 \) into \( P(x) \) to find the remainder.

### 4. **Determine if \( -2 \) is a zero of \( P(x) = x^4 + 2x^3 + 8x - 2 \).**
   - Substitute \( x = -2 \) into \( P(x) \). If the result is zero, then \( -2 \) is a root.

Do you want detailed step-by-step solutions for these problems?

Here are 5 related questions:
1. How does synthetic division work for polynomials?
2. What is the significance of the remainder theorem in polynomial division?
3. How can we verify if a polynomial is divisible by a binomial like \( x - c \)?
4. Can we use the factor theorem for higher degree polynomials? How?
5. How do you interpret the result when dividing two polynomials?

**Tip:** Always check for possible common factors before performing division to simplify the process.

Find the factors of 4x^3 - 10x^2 - 3x + 9, factor the polynomial x^3 - 3x^2 - 5x + 15, find the remainder when dividing a polynomial by x - 4, and determine if -2 is a zero of a given polynomial.

Learn how to factor polynomials using the remainder and factor theorems with detailed examples. This guide covers polynomials like 4x^3 - 10x^2 - 3x + 9 and x^3 - 3x^2 - 5x + 15, and how to determine remainders when dividing by x - 4. Suitable for Grades 10-12.

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