Hoe schrijf je de gedeeltelijke fractie-decompositie van de rationele expressie x ^ 2 / ((x-1) (x + 2))?

Antwoord:

# X ^ 2 / ((x-1) (x + 2)) = 1 / (3 (x-1)) - 4 / (3 (x + 2)) #

Uitleg:

We moeten deze in termen van elke factor schrijven.

# X ^ 2 / ((x-1) (x + 2)) = A / (x-1) + B / (x + 2) #

# X ^ 2 = A (x + 2) + B (x-1) #

Erin stoppen # X = -2 #:

# (- 2) ^ 2 = A (-2 + 2) + B (-2-1) #

# 4 = -3b #

# B = -4/3 #

Erin stoppen # X = 1 #:

# 1 ^ 2 = A (1 + 2) + B (1-1) #

# 1 = 3A #

# A = 1/3 #

# X ^ 2 / ((x-1) (x + 2)) = (1/3) / (x-1) + (- 03/04) / (x + 2) #

#color (wit) (x ^ 2 / ((x-1) (x + 2))) = 1 / (3 (x-1)) - 4 / (3 (x + 2)) #

Antwoord:

# 1 + 1/3 * 1 / (x-1) -4/3 * 1 / (x + 2) #

Uitleg:

# X ^ 2 / (x-1) (x + 2) #

=# (X-1) (x + 2) + x ^ 2- (x-1) (x + 2) / (x-1) (x + 2) #

=# 1 - (x-1) (x + 2) -x ^ 2 / (x-1) (x + 2) #

=# 1- (x-2) / (x-1) (x + 2) #

Nu heb ik breuken in basale gebroken, # (X-2) / (x-1) (x + 2) = A / (x-1) + B / (x + 2) #

Na het uitbreiden van de noemer, # A * (x + 2) + B * (x-1) = x-2 #

set # X = -2 #, # -3b = -4 #, dus # B = 4/3 #

set # X = 1 #, # 3A = -1 #, dus # A = -1/3 #

Vandaar,

# (X-2) / (x-1) (x + 2) = - 1/3 * 1 / (x-1) + 4/3 * 1 / (x + 2) #

Dus, # X ^ 2 / (x-1) (x + 2) #

=# 1- (x-2) / (x-1) (x + 2) #

=# 1 + 1/3 * 1 / (x-1) -4/3 * 1 / (x + 2) #