PreCalc Polynomials

Question

Two polynomials P and D are given. Use either synthetic or long division to divide P(x)

 by D(x),

 and express P in the form P(x) = D(x) · Q(x) + R(x).

P(x) = 8x⁴ + 4x³ + 6x²,    D(x) = 2x² + 1

P(x) =

Answer 1

Long division:

Check:

(2x² + 1)(4x² + 2x + 1) = 8x⁴ + 4x³ + 2x² + 4x² + 2x + 1 = 8x⁴ + 4x³ + 6x² + 2x + 1 then add in the remainder: 8x⁴ + 4x³ + 6x² + 2x + 1 - 2x - 1 = 8x⁴ + 4x³ + 6x² so check.

That means for P(x) = 8x⁴ + 4x³ + 6x² and D(x) = 2x² + 1 that Q(x) = 4x² + 2x + 1 and R(x) = -2x - 1 or:

P(x) = D(x) • Q(x) + R(x)

8x⁴ + 4x³ + 6x² = (2x² + 1)(4x² + 2x + 1) + (-2x - 1)