The remainder when `4x^(6)-5x^(3)-3` is divided by `x^(3)-2` is:

Options

A. 1
B. 0
C. 3
D. 2

Answer

To find the remainder when \( 4x^6 - 5x^3 - 3 \) is divided by \( x^3 - 2 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We are dividing \( 4x^6 - 5x^3 - 3 \) by \( x^3 - 2 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 4x^6 \) by the leading term of the divisor \( x^3 \):
\[ \frac{4x^6}{x^3} = 4x^3 \]
This means we will multiply the entire divisor \( x^3 - 2 \) by \( 4x^3 \).

### Step 3: Multiply and subtract
Now, multiply \( 4x^3 \) by \( x^3 - 2 \):
\[ 4x^3 \cdot (x^3 - 2) = 4x^6 - 8x^3 \]
Now subtract this from the original polynomial:
\[ (4x^6 - 5x^3 - 3) - (4x^6 - 8x^3) = -5x^3 + 8x^3 - 3 = 3x^3 - 3 \]

### Step 4: Repeat the process
Now we will divide \( 3x^3 - 3 \) by \( x^3 - 2 \). Divide the leading term:
\[ \frac{3x^3}{x^3} = 3 \]
Multiply the divisor by 3:
\[ 3 \cdot (x^3 - 2) = 3x^3 - 6 \]
Now subtract this from \( 3x^3 - 3 \):
\[ (3x^3 - 3) - (3x^3 - 6) = -3 + 6 = 3 \]

### Step 5: Conclusion
The remainder when \( 4x^6 - 5x^3 - 3 \) is divided by \( x^3 - 2 \) is \( 3 \).

Thus, the final answer is:
\[ \text{Remainder} = 3 \]