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).

Question

P(x) = 6x^4 + 4x^3 + 5x^2, D(x) = 2x^2 + 1

Answer 1

In order to divide polynomials you need to first put placeholders in for all possible exponent degrees so we'll change P(x) to be:

P(x) = 6x⁴ + 4x³ + 5x² + 0x + 0

And we'll change D(x) to be:

D(x) = 2x² + 0x + 1

Now, set it up like any long division problem:

1) Divide 2x² into 6x⁴ to get 3x². and put the 3x² above the 6x4 as shown, Then multiply 3x² by each of the terms in the dividend (2x², 0x, and 1) and place the results below each term and subtract as indicated:

Keep repeating the process to the end like shown.

The result (quotient) is 3x² + 2x + 1 with remainder -2x - 1

So Q(x) = 3x² + 2x + 1 and R(x) = -2x - 1