Class 14
If there is \((ax + b)^k\) factor in the denominator, the corresponding partial fractions are
\[\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \frac{A_3}{(ax+b)^3} + \cdots + \frac{A_k}{(ax+b)^k}\]
If there is an irreducible \((ax^2 + b)^k\) factor in the denominator, the corresponding partial fractions are
\[\frac{B_1x + C_1}{ax + b} + \frac{B_2 x + C_2}{(ax + b)^2} + \frac{B_3 x + C_3}{(ax+b)^3} + \cdots + \frac{B_k x + C_k}{(ax+b)^k}\]
\(\displaystyle \frac{x^2 + 2x+1}{(x+1)^3(x^2+2)^2}\)
If a polynomial
\[a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1 x + a_0\]
has a rational zero \(r = \frac{p}{q}\) (simplified)
then \(p\) is a factor of \(a_0\) and \(q\) is a factor of \(a_n\).
\[6 x^{5} + 5 x^{4} - 35 x^{3} - 15 x^{2} + 29 x + 10\]
Using Synthetic division (a.k.a nested evaluation a.k.a. H"orner’s algorithm)
\[6 x^{5} + 5 x^{4} - 35 x^{3} - 15 x^{2} + 29 x + 10\]
Factor the polynomial \(2x^4 - 7x^3 - 18x^2 + 13x + 10\).
Write this rational function as a sum of rational functions with prime denominators: \[\frac{x^3 + 3x^2 - 2x + 1}{2x^4 - 7x^3 - 18x^2 + 13x + 10}\]
Factor the polynomial \(3x^4 - x^3 - 9x^2 + 9x - 2\).
\(\displaystyle \int\frac{3 x^{3} + 15 x^{2} - 11 x + 5}{3 x^{4} - x^{3} - 9 x^{2} + 9 x - 2}\;dx\)
\(\displaystyle \int \frac{3x + 1}{x^2 + 2}\;dx\)
Factor the polynomial \(x^4 + 3x^3 + 2x^2 - 2x - 4\).
\(\displaystyle \int \frac{2 x^{3} + 8 x^2 + 5x}{x^{4} + 3 x^{3} + 2 x^{2} - 2 x - 4}\;dx\)