MA42 PFD

Proper vs Improper

100

deg(P) = deg(Q)

Improper

200

deg(P) > deg(Q)

Improper

300

deg(P) < deg(Q)

Proper

400

(ex²) / (-ex³)

Proper

Forming Rules

100

ax + b

A / (ax + b)

200

(ax + b)^k

A/(ax+b) + B/(ax+b)² + ... + #/(ax+b)^k

300

ax² + bx + c

(Ax + B) / (ax² + bx + c)

400

(ax² + bx + c)^k

[(Ax + B) / (ax² + bx + c)] + ... + [(#x + #) / (ax² + bx + c)^k]

Forming Examples

100

(10x) / (x-3)(x+7)

A/(x-3) + B/(x+7)

200

(x-7)/(x²-5x+13)²

(Ax+B)/(x²-5x+13) + (Cx+D)/(x²-5x+13)²

300

(2x²-7) / x⁴

A/x + B/x² + C/x³ + D/x⁴

400

1/[(ex²)(x-2)²]

1/e [A/x + B/x² + C/(x-2) + D/(x-2)²]

Factoring

100

12x³+18x

6x(2x²+3)

200

x²-9

(x+3)(x-3)

300

x²+5x+6

(x+2)(x+3)

400

4x⁴+2x²

2x²(2x²+1)

Misc.

100

integral of: 1/(x-2) dx

ln(|x-2|) + C

200

how can we use PFD for improper functions?

perform polynomial long division to get a reminder plus a proper faction and then do PFD on the proper part

300

form 1/x² in two different ways

A/x + B/x² since x² = (x+0)² and (Ax+B)/x²  

400

integral of: 1/(a² + x²) dx

1/a * arctan(x/a) + C