partial fraction

Question

Clear denominators in the following partial fraction decomposition and determine the constant 𝐵. 9𝑥^2+81𝑥+108 / (𝑥+8)(𝑥+2)^2 =1/(𝑥+8) − 𝐵 /(𝑥+2) − 3/(𝑥+2)^2

Answer 1

B=-8

1/(x+8) - -8/(x+2) - 3/(x+2)^2

rewrite with a common denominator

= (x+2)^2(x+8)(x+2)^2 +8(x+8)(x+2)/(x+8)(x+2)^2 - 3(x+8)/(x+8)(x+2)^2

add the new numerators

= (x^2 +4x +4 + 8x^2 +80x +128 -3x-24)/(x+8)(x+2)^2

combine like terms

=(9x^2 + 81x +108)/(x+8)(x+2)^2

=(9x^2+81x+108)/(x+8)(x^2+4x+4)

= (9x^2+81x+108)/(x^3+12x^2+36x+32)

= A/(x+8) - B/(x+2) - 3/(x+2)^2

=A(x+2)^2 -(B)(x+8)(x+2)-3(x+8)

=Ax^2 +4Ax +4A -Bx^2 -10Bx -16B -3x -24

= (A-B)x^2 +(4A-10B-3)x + 4A-16B-24

A-B=9

A=1, B=1-9=-8