Which of the following options shows the quotient and remainder when `8x^2 - 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 = 4x + 2
C. Quotient = `8x^2 - 10x + 5`, Remainder = 18
D. Quotient = `4x^2 - 6x + 10` , Remainder = 15

Answer

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

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

### Step 2: Divide the leading term
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.

**Hint:** Always start by dividing the leading term of the dividend by the leading term of the divisor.

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

**Hint:** When subtracting, be careful to change the signs of the terms you are subtracting.

### Step 4: Repeat the process
Now, divide the leading term \( 12x^2 \) by \( x \):
\[ \frac{12x^2}{x} = 12x \]
This is the next term of the quotient.

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

**Hint:** Keep repeating the process until you can no longer divide.

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

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

**Hint:** The last subtraction will give you the remainder.

### Conclusion
The quotient is:
\[ 8x^2 + 12x + 25 \]
The remainder is:
\[ 47 \]

Thus, when \( 8x^3 - 4x^2 + x - 3 \) is divided by \( x - 2 \), the quotient is \( 8x^2 + 12x + 25 \) and the remainder is \( 47 \).

### Final Answer
Quotient: \( 8x^2 + 12x + 25 \)
Remainder: \( 47 \)