Use the remainder theorem to find $$ P(1) $$ for $$ P(x)=-2 x^{4}+4 x^{3}-4 x+3 $$. Specifically, give the quotient and the remainder for the associated division and the value of $$ P(1) $$. Quotient $$ =\square $$ Remainder $$ =\square $$ $$ P(1)=\square $$

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To find $$ P(1) $$ using the Remainder Theorem, we will evaluate the polynomial $$ P(x) = -2x^4 + 4x^3 - 4x + 3 $$ at $$ x = 1 $$.

The Remainder Theorem states that the remainder of the division of a polynomial $$ P(x) $$ by $$ x - c $$ is equal to $$ P(c) $$. In this case, we will find $$ P(1) $$.

Let's calculate $$ P(1) $$ step by step:

1. Substitute $$ x = 1 $$ into the polynomial:
   $$
   P(1) = -2(1)^4 + 4(1)^3 - 4(1) + 3
   $$

2. Calculate each term:
   - $$ -2(1)^4 = -2 $$
   - $$ 4(1)^3 = 4 $$
   - $$ -4(1) = -4 $$
   - The constant term is $$ 3 $$

3. Combine the results:
   $$
   P(1) = -2 + 4 - 4 + 3
   $$

Now, let's perform the calculations to find the final value of $$ P(1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$-2+4-4+3$$
- step1: Simplify:
  $$-2+3$$
- step2: Add the numbers:
  $$1$$
Now, let's summarize the results:

1. The value of $$ P(1) $$ is $$ 1 $$.

Next, we need to find the quotient and the remainder when dividing $$ P(x) $$ by $$ x - 1 $$.

Let's perform the polynomial long division of $$ P(x) $$ by $$ x - 1 $$ to find the quotient and the remainder.

The polynomial division will be set up as follows:

- Dividend: $$ -2x^4 + 4x^3 - 4x + 3 $$
- Divisor: $$ x - 1 $$

Let's perform the division.
It seems that the direct approach to find the quotient $$ Q(x) $$ using the equation did not yield a solution. Instead, I will perform polynomial long division manually to find the quotient and the remainder.

### Polynomial Long Division Steps:

1. **Divide the leading term**: 
   - Divide $$ -2x^4 $$ by $$ x $$ to get $$ -2x^3 $$.

2. **Multiply and subtract**:
   - Multiply $$ -2x^3 $$ by $$ (x - 1) $$ to get $$ -2x^4 + 2x^3 $$.
   - Subtract this from the original polynomial:
     $$
     (-2x^4 + 4x^3 - 4x + 3) - (-2x^4 + 2x^3) = 2x^3 - 4x + 3
     $$

3. **Repeat the process**:
   - Divide $$ 2x^3 $$ by $$ x $$ to get $$ 2x^2 $$.
   - Multiply $$ 2x^2 $$ by $$ (x - 1) $$ to get $$ 2x^3 - 2x^2 $$.
   - Subtract:
     $$
     (2x^3 - 4x + 3) - (2x^3 - 2x^2) = 2x^2 - 4x + 3
     $$

4. **Continue**:
   - Divide $$ 2x^2 $$ by $$ x $$ to get $$ 2x $$.
   - Multiply $$ 2x $$ by $$ (x - 1) $$ to get $$ 2x^2 - 2x $$.
   - Subtract:
     $$
     (2x^2 - 4x + 3) - (2x^2 - 2x) = -2x + 3
     $$

5. **Final steps**:
   - Divide $$ -2x $$ by $$ x $$ to get $$ -2 $$.
   - Multiply $$ -2 $$ by $$ (x - 1) $$ to get $$ -2x + 2 $$.
   - Subtract:
     $$
     (-2x + 3) - (-2x + 2) = 1
     $$

### Summary of Results:
- **Quotient**: $$ -2x^3 + 2x^2 + 2x - 2 $$
- **Remainder**: $$ 1 $$
- **Value of $$ P(1) $$**: $$ 1 $$

Now, we can fill in the boxes:

- Quotient $$ = -2x^3 + 2x^2 + 2x - 2 $$
- Remainder $$ = 1 $$
- $$ P(1) = 1 $$
Quotient $$ = -2x^3 + 2x^2 + 2x - 2 $$ Remainder $$ = 1 $$ $$ P(1) = 1 $$

Use the remainder theorem to find for .
Specifically, give the quotient and the remainder for the associated division and the value of .
Quotient
Remainder

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Solution

Polynomial Long Division Steps:

Summary of Results:

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