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## 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.

## Additional Notes

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

$(ax^2+bx+c)^n\leadsto\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\frac{A_3x+B_3}{(ax^2+bx+c)^3}+\cdots+\frac{A_nx+B_n}{(ax^2+bx+c)^n}$.

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