Rational Fractions, Proper/Improper Fractions and Irreducible Polynomials

Whenever we write one polynomial divided by another, say $\dfrac{f(x)}{g(x)}$ with $g(x) \neq 0$, we call the expression a rational fraction (or just a fraction, or a polynomial fraction). For example $\dfrac{3x+2}{x^2-1}$ and $\dfrac{x^4+1}{x^2+3}$ are both rational fractions.

Proper vs. improper. A rational fraction $\dfrac{f(x)}{g(x)}$ is called proper if the degree of $f(x)$ is strictly less than the degree of $g(x)$; otherwise it is called improper. So $\dfrac{3x+2}{x^2-1}$ is proper (degree $1 <$ degree $2$), while $\dfrac{x^4+1}{x^2+3}$ is improper (degree $4 \ge$ degree $2$). This distinction matters because the whole partial-fraction machinery below is built to decompose proper fractions — an improper one first has to be reduced to a polynomial plus a proper remainder fraction (covered in the final section of this chapter).

Irreducible polynomials. A polynomial $p(x)$ is called irreducible (over the reals) if it cannot be written as a product of two real polynomials that are each of lower degree than $p(x)$. Two facts we will use constantly:

• Every linear polynomial $ax+b$ ($a \neq 0$) is automatically irreducible — you can't split a degree-1 polynomial into two factors of even lower degree.
• A quadratic $ax^2+bx+c$ (with $a \neq 0$) is irreducible exactly when its discriminant is negative, i.e. $b^2 - 4ac < 0$. This is precisely the condition under which the quadratic has no real roots, so it can't be factored into two real linear pieces. For instance, $x^2+x+1$ is irreducible since $1 - 4 = -3 < 0$, but $x^2 - 5x + 6 = (x-2)(x-3)$ is reducible. …