Partial Fractions

An expression of the form:

ax+b (px+q)(rx+s)

An expression of the form:

ax+b (px+q)(rx+s) Can be split into partial fractions

A B + (px+q) rx+s

ax+b A B = + (px+q)(rx+s) (px+q) rx+s 13x−6 3x2 −2x

ax+b A B = + (px+q)(rx+s) (px+q) rx+s 13x−6 = 13x−6 3x2 −2x x(3x−2)

ax+b A B = + (px+q)(rx+s) (px+q) rx+s 13x−6 = A B + x(3x−2) x 3x−2

ax+b A B + (px+q)(rx+s) (px+q) rx+s A B = A(3x−2)+Bx = + x 3x−2 x(3x−2)

ax+b A B + (px+q)(rx+s) (px+q) rx+s A B = + x 3x−2 A(3x−2)+Bx x(3x−2)

13x−6 = x(3x−2)

ax+b A B + (px+q)(rx+s) (px+q) rx+s A B = 13x−6 = + x(3x−2) x 3x−2 A(3x−2)+Bx x(3x−2)

3Ax−2A+Bx = x(3x−2)

A B 13x−6 = + = x 3x−2 x(3x−2) A(3x−2)+Bx 3Ax−2A+Bx = x(3x−2) x(3x−2)

A B 13x−6 = + = x 3x−2 x(3x−2) A(3x−2)+Bx 3Ax−2A+Bx = x(3x−2) x(3x−2)

Equating coefficients of x

3A + B = 13

13x−6 3Ax−2A+Bx = x(3x−2) = x(3x−2) Equating coefficients of x

3A + B = 13 −2A

= −6

−6 A = −2

A B 13x−6 = + = x 3x−2 x(3x−2) Equating coefficients of x

3A + B = 13 −2A

= −6

−6 A = −2

A=3

A B 13x−6 = + = x 3x−2 x(3x−2) Equating coefficients of x

3A + B = 13 3(3) + B = 13

−6 A = −2

A=3

A B 13x−6 = + = x 3x−2 x(3x−2) Equating coefficients of x

3A + B = 13 3(3) + B = 13 B = 13 − 9

−6 A = −2

A=3

A B 13x−6 = + = x 3x−2 x(3x−2) Equating coefficients of x

3A + B = 13 3(3) + B = 13 B = 13 − 9 B=4

−6 A = −2

A=3

A B 13x−6 + = x 3x−2 x(3x−2) 13x−6 3 4 = + x(3x−2) 3x−2 x

12x (x+1)(2x+3)(x−3)