I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

In Week 3 we looked at Partial Fractions. They provide us with a way of integrating rational functions and will be very important later for when we study Laplace Methods.

## Next Week

We will look start Multivariable Calculus and perhaps begin to look at its application to error analysis — very important for the analysis of physics experiments/data. We will do some Partial Fractions & Multivariable Calculus in our Maple lab.

I skipped over some of the more complicated Partial Fractions stuff that we don’t need for exams. For completeness I have put it here.

### Examples

By dividing $x-1$ into $x^3+x$, write $\frac{x^3+x}{x-1}$

in the same form as (1.5)

Solution:  Do $(x^3+0x^2+x+0)\div (x-1)$, polynomial long division like the example on page 40. It will go in $x^2+x+2$ times with a remainder of $2$. Now compare $17\div 3$. Three goes into 17 five times with a remainder of two and we write: $\frac{17}{3}=5+\frac{2}{3}$.

Similarly we write $\frac{x^3+x}{x-1}=x^2+x+2+\frac{2}{x-1}$.

Write the following in the same form as (1.5) $f(x)=\frac{x^4+3x^2-2}{x^2+1}$.

Solution: Do $(x^4+0x^3+3x^2+0x-2)\div (x^2+0x+1)$ polynomial long division. You will get $x^2+2$ with a remainder of $-4$ so we have $f(x)=x^2+2-\frac{4}{x^2+1}$.

### Rule II: Extension to cubes, fourth powers, etc

To each linear factor of the form $(ax+b)^n$ (i.e. a repeated linear factor of $q(x)$), there corresponds a sum of $n$ partial fraction terms of the form $(ax+b)^n\leadsto \frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+\frac{A_3}{(ax+b)^3}+\cdots+\frac{A_n}{(ax+b)^n}$.

### Rule IV

To each quadratic factor of the form $(ax^2+bx+c)^n$ (i.e. a repeated quadratic factor of $q(x)$), there corresponds a sum of $n$ partial fraction terms of the form $(ax^2+bx+c)^n\leadsto\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\frac{A_3x+B_3}{(ax^2+bx+c)^3}+\cdots+\frac{A_nx+B_n}{(ax^2+bx+c)^n}$.