Find the remainder is : When `x^(51)` is divided by `x^(2)-3x+2`.

Options

A. x
B. `(2^(51)-2)x=2-2^(51)`
C. `(2^(51)-1)x+2-2^(51)`
D. 0

Answer

To find the remainder when \( x^{51} \) is divided by \( x^2 - 3x + 2 \), we can follow these steps:

### Step 1: Identify the polynomials
Let:
- \( P(x) = x^{51} \)
- \( G(x) = x^2 - 3x + 2 \)

### Step 2: Factor the divisor polynomial
We can factor \( G(x) \):
\[ G(x) = (x - 1)(x - 2) \]
This shows that the roots of the polynomial \( G(x) \) are \( x = 1 \) and \( x = 2 \).

### Step 3: Use the Remainder Theorem
According to the Remainder Theorem, the remainder \( R(x) \) when dividing \( P(x) \) by \( G(x) \) will be a linear polynomial of the form:
\[ R(x) = ax + b \]
where \( a \) and \( b \) are constants that we need to determine.

### Step 4: Set up equations using the roots
We will substitute the roots of \( G(x) \) into \( P(x) \) to create a system of equations.

1. Substitute \( x = 1 \):
\[ P(1) = 1^{51} = 1 \]
\[ R(1) = a(1) + b = a + b \]
This gives us the equation:
\[ a + b = 1 \quad \text{(Equation 1)} \]

2. Substitute \( x = 2 \):
\[ P(2) = 2^{51} \]
\[ R(2) = a(2) + b = 2a + b \]
This gives us the equation:
\[ 2a + b = 2^{51} \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
We now have a system of two equations:
1. \( a + b = 1 \)
2. \( 2a + b = 2^{51} \)

Subtract Equation 1 from Equation 2:
\[ (2a + b) - (a + b) = 2^{51} - 1 \]
This simplifies to:
\[ a = 2^{51} - 1 \]

Now substitute \( a \) back into Equation 1:
\[ (2^{51} - 1) + b = 1 \]
\[ b = 1 - (2^{51} - 1) = 2 - 2^{51} \]

### Step 6: Write the remainder
Now we have:
\[ a = 2^{51} - 1 \quad \text{and} \quad b = 2 - 2^{51} \]
Thus, the remainder \( R(x) \) is:
\[ R(x) = (2^{51} - 1)x + (2 - 2^{51}) \]

### Final Answer
The final remainder when \( x^{51} \) is divided by \( x^2 - 3x + 2 \) is:
\[ R(x) = (2^{51} - 1)x + (2 - 2^{51}) \]