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

Question

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We are given  
$$
P(x)=2x^3-3x^2-x+4
$$  
and we want to find the quotient and remainder when $$ P(x) $$ is divided by $$(x-1)$$, and then use the remainder theorem to evaluate $$ P(1) $$.

### Step 1. Recall the Remainder Theorem  
The remainder theorem states that when a polynomial $$ P(x) $$ is divided by $$(x-c)$$, the remainder is $$ P(c) $$. Here, $$ c=1 $$.

### Step 2. Perform Synthetic Division  
We set up synthetic division with the value $$ 1 $$ and the coefficients of $$ P(x) $$:  
$$
\begin{array}{r|cccc}
1 & 2 & -3 & -1 & 4 \
  &   & 2  & -1 & -2 \ \hline
  & 2 & -1 & -2 & 2
\end{array}
$$

**Explanation:**
1. Bring down the first coefficient: $$ 2 $$.
2. Multiply $$ 2 $$ by $$ 1 $$ to get $$ 2 $$ and add to the second coefficient $$ -3 $$ to get: $$ -3+2=-1 $$.
3. Multiply $$ -1 $$ by $$ 1 $$ to get $$ -1 $$ and add to the third coefficient $$ -1 $$ to get: $$ -1+(-1)=-2 $$.
4. Multiply $$ -2 $$ by $$ 1 $$ to get $$ -2 $$ and add to the last coefficient $$ 4 $$ to get: $$ 4+(-2)=2 $$.

The last number, $$ 2 $$, is the remainder.

### Step 3. Identify the Quotient and Remainder  
The numbers $$ 2, -1, -2 $$ (from left to right) are the coefficients of the quotient polynomial, which is of degree 2. Thus, the quotient is:  
$$
2x^2 - x - 2
$$  
and the remainder is:  
$$
2
$$

### Step 4. Evaluate $$ P(1) $$  
By the remainder theorem,  
$$
P(1)=\text{Remainder}=2
$$

### Final Answers  
Quotient $$ =2x^2 - x - 2 $$  
Remainder $$ =2 $$  
$$ P(1)=2 $$
Quotient $$ =2x^2 - x - 2 $$ Remainder $$ =2 $$ $$ P(1)=2 $$

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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Answer

Solution

Step 1. Recall the Remainder Theorem

Step 2. Perform Synthetic Division

Step 3. Identify the Quotient and Remainder

Step 4. Evaluate

Final Answers

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