Use the remainder theorem to find P(2) for P(x)=2x^4-2x^3-6x^2+4. specifically give the quotient and the remainder for the associated division and the value of P(2) 

The Remainder Theorem states that for a polynomial \( P(x) \), the remainder when \( P(x) \) is divided by \( x - c \) is equal to \( P(c) \). So to find \( P(2) \) for \( P(x) = 2x^4 - 2x^3 - 6x^2 + 4 \), we can perform the division of \( P(x) \) by \( x - 2 \).

### Step 1: Perform Polynomial Division

We will divide \( P(x) = 2x^4 - 2x^3 - 6x^2 + 4 \) by \( x - 2 \).

1. **Divide the leading term** of \( 2x^4 \) by the leading term of \( x - 2 \):
   \[
   \frac{2x^4}{x} = 2x^3
   \]
   This gives us the first term of the quotient: \( 2x^3 \).

2. **Multiply** \( 2x^3 \) by \( x - 2 \):
   \[
   2x^3 \times (x - 2) = 2x^4 - 4x^3
   \]
   
3. **Subtract** this result from the original polynomial:
   \[
   (2x^4 - 2x^3 - 6x^2 + 4) - (2x^4 - 4x^3) = 2x^3 - 6x^2 + 4
   \]

4. **Repeat the process**: Divide \( 2x^3 \) by \( x \):
   \[
   \frac{2x^3}{x} = 2x^2
   \]
   This gives us the next term of the quotient: \( 2x^2 \).

5. **Multiply** \( 2x^2 \) by \( x - 2 \):
   \[
   2x^2 \times (x - 2) = 2x^3 - 4x^2
   \]

6. **Subtract** this from the current polynomial:
   \[
   (2x^3 - 6x^2 + 4) - (2x^3 - 4x^2) = -2x^2 + 4
   \]

7. **Repeat the process**: Divide \( -2x^2 \) by \( x \):
   \[
   \frac{-2x^2}{x} = -2x
   \]
   This gives us the next term of the quotient: \( -2x \).

8. **Multiply** \( -2x \) by \( x - 2 \):
   \[
   -2x \times (x - 2) = -2x^2 + 4x
   \]

9. **Subtract**:
   \[
   (-2x^2 + 4) - (-2x^2 + 4x) = -4x + 4
   \]

10. **Repeat the process**: Divide \( -4x \) by \( x \):
    \[
    \frac{-4x}{x} = -4
    \]
    This gives us the last term of the quotient: \( -4 \).

11. **Multiply** \( -4 \) by \( x - 2 \):
    \[
    -4 \times (x - 2) = -4x + 8
    \]

12. **Subtract**:
    \[
    (-4x + 4) - (-4x + 8) = -4
    \]

Thus, the quotient is \( 2x^3 + 2x^2 - 2x - 4 \) and the remainder is \( -4 \).

### Step 2: Find \( P(2) \)

According to the Remainder Theorem:
\[
P(2) = \text{remainder} = -4
\]

### Summary:
- **Quotient**: \( 2x^3 + 2x^2 - 2x - 4 \)
- **Remainder**: \( -4 \)
- **Value of \( P(2) \)**: \( -4 \)

Would you like more details or have any questions?  

Here are some related questions:
1. How would you use synthetic division to find \( P(2) \)?
2. What is the general form of the Remainder Theorem?
3. How does the degree of the polynomial affect the quotient?
4. Can the remainder ever be higher in degree than the divisor?
5. What happens if \( P(x) \) is divisible by \( x - c \)?
6. How does this method apply if \( c \) is a fraction?
7. What is the Factor Theorem and how is it related?
8. Can you find the quotient and remainder for \( P(x) = 3x^3 - x^2 + x - 5 \) divided by \( x - 1 \)?

**Tip:** Remember that the Remainder Theorem not only gives you the remainder but also provides a quick way to evaluate polynomials at a given point.

Learn how to apply the Remainder Theorem to find P(2) for the polynomial P(x) = 2x^4 - 2x^3 - 6x^2 + 4. Understand the quotient and remainder obtained from the division process. Suitable for high school students studying polynomial functions and the Remainder Theorem.

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