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To find the remainder of \( f(x) = x^4 - 3x^3 - 2x^2 - 6x - 4 \) when it is divided by \( x + 2 \), we'll use synthetic division and the Remainder Theorem. The Remainder Theorem states that the remainder of \( f(x) \) divided by \( x - c \) is simply \( f(c) \). Here, \( c = -2 \).

### Step-by-Step Solution:

1. **Identify the divisor**: \( x + 2 \) implies \( c = -2 \).
2. **Write the coefficients** of \( f(x) \): \( [1, -3, -2, -6, -4] \).
3. **Set up synthetic division** with \( c = -2 \):

   | Coefficients: | \( 1 \) | \( -3 \) | \( -2 \) | \( -6 \) | \( -4 \) |
   |---------------|---------|----------|----------|----------|----------|
   | Multiply & Add|         | \( -2 \) | \( 10 \) | \( -16 \)| \( 32 \) |
   | New Row:      | \( 1 \) | \( -5 \) | \( 8 \)  | \( -22 \)| \( 28 \) |

4. The last value in the synthetic division row is the **remainder**: \( 28 \).

Therefore, the remainder when \( f(x) \) is divided by \( x + 2 \) is **28**.

Would you like further details or have any questions? Here are some related queries:

1. How is synthetic division different from long division in polynomials?
2. What does the Remainder Theorem tell us in polynomial functions?
3. Can we use synthetic division if the divisor isn’t in the form \( x - c \)?
4. How would the remainder change if the divisor were \( x - 2 \) instead?
5. How can synthetic division help in finding factors of a polynomial?

**Tip:** When using synthetic division, always ensure the divisor is in the form \( x - c \) and that the polynomial has all powers of \( x \) represented, using coefficients of zero where necessary.

Use synthetic division and the remainder theorem to find the remainder when f(x) is divided by x + 2. Given f(x) = x^4 - 3x^3 - 2x^2 - 6x - 4.

Learn how to find the remainder when dividing the polynomial f(x) = x^4 - 3x^3 - 2x^2 - 6x - 4 by x + 2. This solution uses synthetic division and the remainder theorem, making it ideal for high school algebra students in Grades 10-12.

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