1219.sagews

EXAMPLE 1  Find the partial fraction decomposition of $\frac{x + 14}{(x - 4)(x + 2)}$.

var('A B')

(x+14)/((x-4)*(x+2)) == A/(x-4) + B/(x+2)

expand(-((x-4)*B-x-14)/(x+2))

expand(A)

x+14 == A*(x+2) + B*(x-4)

expand(_)

x+14 == (A+B)*x + 2*(A-2*B)

solve([A+B == 1, 2*A-4*B == 14], [A, B])

(x+14)/((x-4)*(x+2)) == 3/(x-4) - 2/(x+2)

EXAMPLE 2  Find the partial fraction decomposition of $\frac{x - 18}{x(x-3)^2}$.

var('A B C')

(x-18)/(x*(x-3)^2) == A/x + B/(x-3) + C/(x-3)^2

(x-18) == A*(x-3)^2 + B*x*(x-3) + C*x

x - 18 == expand(A*(x-3)^2 + B*x*(x-3) + C*x)

solve([A+B == 0, -6*A-3*B+C == 1, 9*A == -18], [A, B, C])

A, B, C = -2, 2, -5

(x-18)/(x*(x-3)^2) == A/x + B/(x-3) + C/(x-3)^2

EXAMPLE 3  Find the partial fraction decomposition of $\frac{3x^2 + 17x + 14}{(x - 2)(x^2 + 2x + 4)}$.

var('A B C')

(3*x^2+17*x+14)/((x-2)*(x^2+2*x+4)) == A/(x-2) + (B*x+C)/(x^2+2*x+4)

3*x^2 + 17*x + 14 == A*(x^2+2*x+4)+(x-2)*(B*x+C)

3*x^2 + 17*x + 14 == expand(A*(x^2+2*x+4)+(x-2)*(B*x+C))

solve([A+B==3, 2*A-2*B+C==17, 4*A-2*C == 14], [A, B, C])

A, B, C = 5, -2, 3

(3*x^2+17*x+14)/((x-2)*(x^2+2*x+4)) == A/(x-2) + (B*x+C)/(x^2+2*x+4)

EXAMPLE 4  Find the partial fraction decomposition of $\frac{5x^3 - 3x^2 + 7x -3}{(x^2 + 1)^2}$.

var('A B C D')

(5*x^3 - 3*x^2 + 7*x - 3)/(x^2 + 1)^2 == (A*x + B)/(x^2 + 1) + (C*x + D)/(x^2 + 1)^2

5*x^3 - 3*x^2 + 7*x - 3 == (A*x+B)*(x^2+1)+(C*x+D)

5*x^3 - 3*x^2 + 7*x - 3 == expand((A*x+B)*(x^2+1)+(C*x+D))

solve([A == 5, B == -3, A+C == 7, B+D == -3], [A, B, C, D])

A, B, C, D = 5, -3, 2, 0

(5*x^3 - 3*x^2 + 7*x - 3)/(x^2 + 1)^2 == (A*x + B)/(x^2 + 1) + (C*x + D)/(x^2 + 1)^2