Using remainder theorem, find the remainder when `x^(4)+x^(3)-2x^(2)+x+1` is divided by `x-1` .

Answer

To find the remainder when the polynomial \( f(x) = x^4 + x^3 - 2x^2 + x + 1 \) is divided by \( x - 1 \) using the Remainder Theorem, we will follow these steps:

### Step 1: Identify the polynomial and the divisor
We have the polynomial:
\[ f(x) = x^4 + x^3 - 2x^2 + x + 1 \]
and we want to divide it by \( x - 1 \).

### Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, the remainder of the division of \( f(x) \) by \( x - a \) is given by \( f(a) \). In this case, we set \( a = 1 \) because we are dividing by \( x - 1 \).

### Step 3: Calculate \( f(1) \)
Now we will substitute \( x = 1 \) into the polynomial \( f(x) \):
\[ f(1) = (1)^4 + (1)^3 - 2(1)^2 + (1) + 1 \]

### Step 4: Simplify the expression
Now we will simplify the expression step by step:
\[ f(1) = 1 + 1 - 2 + 1 + 1 \]

Calculating this step by step:
1. \( 1 + 1 = 2 \)
2. \( 2 - 2 = 0 \)
3. \( 0 + 1 = 1 \)
4. \( 1 + 1 = 2 \)

Thus, we find:
\[ f(1) = 2 \]

### Step 5: State the remainder
Therefore, the remainder when \( f(x) \) is divided by \( x - 1 \) is:
\[ \text{Remainder} = 2 \]

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