Jika (3x² - x + 2)/[x(x² - 3)] ≡ [(A/x) + (B/(x + 1)) + (C/(x - 1))] nilai A + B + C sama dengan .... A. 0 B. 3 C. 6 D. 5 E. 12

Question

Jawaban : B. 3.

Ingat!
Jika (ax²+bx+c) /d = (ex²+fx+g)/d
Maka a=e, b=f, c=g.

x²-y² =(x-y) (x+y).
(a+b) (c+d) = ac +ad + bc + bd.

Asumsikan (3x² - x + 2)/[x(x² - 1)] ≡ [(A/x) + (B/(x + 1)) + (C/(x - 1))].

Perhatikan.

(3x² - x + 2)/[x(x² - 1)] ≡ [(A/x) + (B/(x + 1)) + (C/(x - 1))]

(3x² - x + 2)/[x(x² - 1)] ≡ [A(x + 1)(x - 1)]/[x(x + 1)(x - 1)] + [Bx(x - 1)]/[x(x + 1)(x - 1)] + [Cx(x + 1)]/[x(x + 1)(x - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [A(x² - 1)]/[x(x² - 1)] + [B(x² - x)]/[x(x² - 1)] + [C(x² + x)]/[x(x² - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [Ax² - A]/[x(x² - 1)] + [Bx² - Bx]/[x(x² - 1)] + [Cx² + Cx]/[x(x² - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [(Ax² - A) + (Bx² - Bx) + (Cx² + Cx)]/[x(x² - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [Ax² - A + Bx² - Bx + Cx² + Cx]/[x(x² - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [Ax² + Bx² + Cx² - Bx + Cx - A]/[x(x² - 1)]

(3x² - x + 2)/[x(x² - 1)] ≡ [(A + B + C)x² - (B - C)x - A]/[x(x² - 1)]

Diperoleh

3x² = (A + B + C)x²

A + B + C = 3 .....(1)

- x = - (B - C)x

B - C = 1 .....(2)

2 = -A

A = -2 .....(3)

Substitusi A = -2 ke A + B + C = 3.

-2 + B + C = 3

B + C = 3+2

B + C = 5 .....(4)

Eliminasi C pada persamaan (2) dan (4)
B - C = 1
B + C = 5
_________+
2B = 6

B = 6/2

B = 3

Substitusi B = 3 ke B - C = 1.
3 - C = 1

-C = 1-3

-C = -2

C = -2/-1

C = 2

Nilai A = -2, B = 3 dan C = 2, sehingga A + B + C = -2 + 3 + 2 = 3.

Jadi, jawaban yang benar adalah B. 3.

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