Question

Find the quotient when p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by (x² + 3x + 1).

Answer 1

Given: p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 ; Divisor = x² + 3x + 1.

Step-wise calculation:

1. Divide leading terms: 3x⁴ ÷ x² = 3x².

Multiply divisor: 3x²(x² + 3x + 1) = 3x⁴ + 9x³ + 3x².

Subtract from p(x): (3x⁴ + 5x³ – 7x²) – (3x⁴ + 9x³ + 3x²) = –4x³ – 10x²

Bring down + 2x → –4x³ – 10x² + 2x.

2. Divide leading terms: (–4x³) ÷ x² = –4x.

Multiply divisor: –4x(x² + 3x + 1) = –4x³ – 12x² – 4x

Subtract: (–4x³ – 10x² + 2x) – (–4x³ – 12x² – 4x) = 2x² + 6x

Bring down + 2 → 2x² + 6x + 2.

3. Divide leading terms: 2x² ÷ x² = 2.

Multiply divisor: 2(x² + 3x + 1) = 2x² + 6x + 2.

Subtract: (2x² + 6x + 2) – (2x² + 6x + 2) = 0 → remainder 0.

Quotient = 3x² – 4x + 2 and remainder 0.