What is the remainder if `p(x)=x^(3)+2x+1` is divided by `x-2` ?

Answer

To find the remainder when \( p(x) = x^3 + 2x + 1 \) is divided by \( x - 2 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( p(x) \) by \( x - c \) is \( p(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
We have the polynomial \( p(x) = x^3 + 2x + 1 \) and we are dividing by \( x - 2 \).

2. **Determine the value of \( c \)**:
Here, \( c = 2 \) (since we are dividing by \( x - 2 \)).

3. **Evaluate the polynomial at \( c \)**:
We need to find \( p(2) \):
\[ p(2) = (2)^3 + 2(2) + 1 \]

4. **Calculate \( p(2) \)**:
\[ p(2) = 8 + 4 + 1 = 13 \]

5. **State the remainder**:
Therefore, the remainder when \( p(x) \) is divided by \( x - 2 \) is \( 13 \).

### Final Answer:
The remainder is \( 13 \).

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