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When you divide 1717 by 55, you get 33 with a remainder of 22, which we write as 3+253 + \frac{2}{5}.
What happens when the numbers turn into polynomials? The exact same logic allows us to break complicated rational expressions into simple, bite-sized components.

Proper vs. Improper Algebraic Fractions

The degree of a polynomial is the highest power of the variable appearing with a non-zero number in front of it.
An improper rational expression has a numerator whose degree is greater than or equal to the degree of its denominator.
By the Polynomial Division Algorithm, any improper fraction can be expressed uniquely as P(x)Q(x)=Q(x)quotient+R(x)Q(x)\frac{P(x)}{Q(x)} = Q(x)_{\text{quotient}} + \frac{R(x)}{Q(x)}, where deg⁡(R(x))<deg⁡(Q(x))\deg(R(x)) < \deg(Q(x)).
How do we find the polynomial quotient Q(x)Q(x) and remainder R(x)R(x) systematically?

Method 1: Polynomial Long Division

Polynomial long division mirrors grade-school long division: divide the highest-degree terms, multiply back, and subtract.
Let's divide P(x)=2x3−3x2+5P(x) = 2x^3 - 3x^2 + 5 by D(x)=x−2D(x) = x - 2. Because the x1x^1 term is absent, we insert a placeholder +0x+ 0x to keep columns aligned.
The division yields a quotient Q(x)=2x2+x+2Q(x) = 2x^2 + x + 2 and a remainder R(x)=9R(x) = 9, meaning 2x3−3x2+5x−2=2x2+x+2+9x−2\frac{2x^3 - 3x^2 + 5}{x - 2} = 2x^2 + x + 2 + \frac{9}{x - 2}.