Which of the following options shows the quotient and remainder when `8x^(3) - 4x^(2) + x - 3` is divisible by x - 2 ?

Options

A. Quotient `= 8x^(2) + 12x + 25` , Remainder = 47
B. Quotient ` = 4x^(2) + 12x - 20`, Remainder `= 4 x + 2`
C. Quotient ` = 8x^(2) - 10x + 5` , Remainder
D. Quotient ` = 4x^(2) - 6x + 10` , Remainder = 15

Answer

To find the quotient and remainder when dividing \(8x^3 - 4x^2 + x - 3\) by \(x - 2\), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We will divide the polynomial \(8x^3 - 4x^2 + x - 3\) by \(x - 2\).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(8x^3\) by the leading term of the divisor \(x\):
\[ \frac{8x^3}{x} = 8x^2 \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(8x^2\) by the entire divisor \(x - 2\):
\[ 8x^2(x - 2) = 8x^3 - 16x^2 \]
Subtract this from the original polynomial:
\[ (8x^3 - 4x^2 + x - 3) - (8x^3 - 16x^2) = (16x^2 - 4x^2) + x - 3 = 12x^2 + x - 3 \]

### Step 4: Repeat the process
Now, repeat the process with the new polynomial \(12x^2 + x - 3\).

Divide the leading term \(12x^2\) by \(x\):
\[ \frac{12x^2}{x} = 12x \]
This is the next term of the quotient.

Multiply \(12x\) by the divisor:
\[ 12x(x - 2) = 12x^2 - 24x \]
Subtract this from \(12x^2 + x - 3\):
\[ (12x^2 + x - 3) - (12x^2 - 24x) = (24x + x) - 3 = 25x - 3 \]

### Step 5: Repeat again
Now, divide \(25x\) by \(x\):
\[ \frac{25x}{x} = 25 \]
This is the next term of the quotient.

Multiply \(25\) by the divisor:
\[ 25(x - 2) = 25x - 50 \]
Subtract this from \(25x - 3\):
\[ (25x - 3) - (25x - 50) = -3 + 50 = 47 \]

### Conclusion
Now we have the final quotient and remainder:
- Quotient: \(8x^2 + 12x + 25\)
- Remainder: \(47\)

Thus, the answer is:
- Quotient: \(8x^2 + 12x + 25\)
- Remainder: \(47\)