Question

Divide polynomial \( 2x^4 + 3x^3 - 2x^2 - 9x - 12 \) by polynomial \( x^2 - 3 \).

Hint:

Polynomial Division: Divide each term successively using \( x^2 - 3 \).

Answer 1

Using polynomial division: \[ \frac{2x^4 + 3x^3 - 2x^2 - 9x - 12}{x^2 - 3} \] 1. Divide \( 2x^4 \) by \( x^2 \): \[ 2x^2 \] Multiply: \[ 2x^4 - 6x^2 \] Subtract: \[ (2x^4 + 3x^3 - 2x^2 - 9x - 12) - (2x^4 - 6x^2) \] \[ 3x^3 + 4x^2 - 9x - 12 \] 2. Divide \( 3x^3 \) by \( x^2 \): \[ 3x \] Multiply: \[ 3x^3 - 9x \] Subtract: \[ (3x^3 + 4x^2 - 9x - 12) - (3x^3 - 9x) \] \[ 4x^2 - 12 \] 3. Divide \( 4x^2 \) by \( x^2 \): \[ 4 \] Multiply: \[ 4x^2 - 12 \] Subtract: \[ (4x^2 - 12) - (4x^2 - 12) = 0 \] Thus, the quotient is: \[ 2x^2 + 3x + 4 \] Correct Answer: \( 2x^2 + 3x + 4 \)