Question

If the polynomial (x⁴ + 2x³ + 8x² + 12x + 18) is divided by another polynomial (x² + 5), the remainder comes out to be (px + q). Find the values of p and q.

Answer 1

Given: Dividend f(x) = x⁴ + 2x³ + 8x² + 12x + 18, divisor g(x) = x² + 5, remainder r(x) = px + q.

Step-wise calculation:

1. If α is a root of g(x) then α² = –5 and f(α) = r(α) = pα + q.

2. Compute f(α) using α² = –5:

α⁴ = (α²)²

= (–5)²

= 25

α³ = α × α²

= α × (–5) 

= –5α 

So, f(α) = α⁴ + 2α³ + 8α² + 12α + 18 

= 25 + 2(–5α) + 8(–5) + 12α + 18

= 25 – 10α – 40 + 12α + 18

= (25 – 40 + 18) + (–10α + 12α)

= 3 + 2α

3. Thus for any root α of g, pα + q = 2α + 3.

Equating coefficients gives p = 2 and q = 3.