division of polynomials of same degree

Question

I am solving a differential equation which requires solving and integrating $(4y^3 + 10y^2 + 5y + 2) / (2y^3 + 3y^2 + 3y + 1)$. I am thinking synthetic division to reduce it but am not sure where to start as both polynomials are of the same degree.

Answer 1

Synthetic division is for dividing by a degree 1 term of form $x-k.$ So that won't do this division. Use "long division" to do it, remainder should have degree less than 3.

Answer 2

When you divide polynomials of the same degree the quotient is a constant and the remainder is usually of one degree less. The quotient comes from the ratio of the leading terms.

Here you have $$\frac{4y^3 + 10y^2 + 5y + 2 }{ 2y^3 + 3y^2 + 3y + 1}=2+\frac{ 4y^2 -y }{ 2y^3 + 3y^2 + 3y + 1}$$

Answer 3

The first step is to divide the two polynomials. For the same degree, you get a constant plus a ratio where the numerator is at least one degree less. In this case, look at @RossMillikan ' s answer. This might be still problematic to integrate, so you look for roots of the denominator. $-1/2$ is a real root. You should then write your fraction as a sum of ratios, one with denominator $x+1/2$, the other with the denominator $(2y^3+3y^2+3y+1)/(x+1/2)$