Kako napišete delni razčlenitev racionalnega izraza x ^ 2 / ((x-1) (x + 2))?

Odgovor:

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

Pojasnilo:

To moramo zapisati v smislu vsakega dejavnika.

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

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

Vstavljanje # x = -2 #:

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

# 4 = -3B #

# B = -4 / 3 #

Vstavljanje # 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) + (- 4/3) / (x + 2) #

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

Odgovor:

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

Pojasnilo:

# 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) #

Zdaj sem razčlenil frakcijo v osnovne, # (x-2) / (x-1) (x + 2) = A / (x-1) + B / (x + 2) #

Po razširitvi imenovalca, # A * (x + 2) + B * (x-1) = x-2 #

Set # x = -2 #, # -3B = -4 #, Torej # B = 4/3 #

Set # x = 1 #, # 3A = -1 #, Torej # A = -1 / 3 #

Zato

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

Tako

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

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

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