What is the result when `4x ^(3) -5x ^(2) +x - 3` is divided by `x - 2` ?

Options

A. `4x +3+ (11)/(x -2)`
B. `4x ^(2) +3x -6`
C. `4x ^(2) + 3x + 18`
D. `4x ^(2) + 3x + 7+ (11)/(x -2)`

Answer

To solve the problem of dividing the polynomial \( 4x^3 - 5x^2 + x - 3 \) by \( x - 2 \), we will use the long division method for polynomials. Here are the steps:

### Step 1: Set Up the Division
We will divide \( 4x^3 - 5x^2 + x - 3 \) by \( x - 2 \).

### Step 2: Divide the Leading Terms
Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( x \):
\[ \frac{4x^3}{x} = 4x^2 \]
This is the first term of our quotient.

### Step 3: Multiply and Subtract
Now, multiply \( 4x^2 \) by \( x - 2 \):
\[ 4x^2 \cdot (x - 2) = 4x^3 - 8x^2 \]
Subtract this from the original polynomial:
\[ (4x^3 - 5x^2 + x - 3) - (4x^3 - 8x^2) = -5x^2 + 8x^2 + x - 3 = 3x^2 + x - 3 \]

### Step 4: Repeat the Process
Now, we will divide the new leading term \( 3x^2 \) by \( x \):
\[ \frac{3x^2}{x} = 3x \]
This is the next term of our quotient.

### Step 5: Multiply and Subtract Again
Multiply \( 3x \) by \( x - 2 \):
\[ 3x \cdot (x - 2) = 3x^2 - 6x \]
Subtract this from \( 3x^2 + x - 3 \):
\[ (3x^2 + x - 3) - (3x^2 - 6x) = x + 6x - 3 = 7x - 3 \]

### Step 6: One More Time
Now, divide \( 7x \) by \( x \):
\[ \frac{7x}{x} = 7 \]
This is the next term of our quotient.

### Step 7: Final Multiply and Subtract
Multiply \( 7 \) by \( x - 2 \):
\[ 7 \cdot (x - 2) = 7x - 14 \]
Subtract this from \( 7x - 3 \):
\[ (7x - 3) - (7x - 14) = -3 + 14 = 11 \]

### Conclusion
Now we have completed the division. The quotient is \( 4x^2 + 3x + 7 \) and the remainder is \( 11 \). Therefore, we can express the result of the division as:
\[ \frac{4x^3 - 5x^2 + x - 3}{x - 2} = 4x^2 + 3x + 7 + \frac{11}{x - 2} \]

### Final Answer
Thus, the result when \( 4x^3 - 5x^2 + x - 3 \) is divided by \( x - 2 \) is:
\[ 4x^2 + 3x + 7 + \frac{11}{x - 2} \]