Jabarkan dan sederhanakan bentuk-bentuk berikut. a.

a. 81a^4 + 108a^3b^2 + 54a^2b^4 + 12ab^6 + b^8 b. 32p^5 + 80p^4q^2 + 80p^3q^4 + 40p^2q^6 + 10pq^8 + q^10 c. 243p^10 - 405p^8q + 270p^6q^2 - 90p^4q^3 + 15p^2q^4 - q^5

Pembahasan

Untuk menjabarkan dan menyederhanakan bentuk-bentuk tersebut, kita akan menggunakan teorema binomial.
a. (3a+b^2)^4
Menggunakan teorema binomial: (x+y)^n = Σ (n choose k) * x^(n-k) * y^k, dimana (n choose k) = n! / (k! * (n-k)!).
Untuk (3a+b^2)^4:
n=4, x=3a, y=b^2
(4 choose 0)(3a)^4(b^2)^0 + (4 choose 1)(3a)^3(b^2)^1 + (4 choose 2)(3a)^2(b^2)^2 + (4 choose 3)(3a)^1(b^2)^3 + (4 choose 4)(3a)^0(b^2)^4
= 1 * (81a^4) * 1 + 4 * (27a^3) * b^2 + 6 * (9a^2) * b^4 + 4 * (3a) * b^6 + 1 * 1 * b^8
= 81a^4 + 108a^3b^2 + 54a^2b^4 + 12ab^6 + b^8
b. (2p+q^2)^5:
n=5, x=2p, y=q^2
(5 choose 0)(2p)^5(q^2)^0 + (5 choose 1)(2p)^4(q^2)^1 + (5 choose 2)(2p)^3(q^2)^2 + (5 choose 3)(2p)^2(q^2)^3 + (5 choose 4)(2p)^1(q^2)^4 + (5 choose 5)(2p)^0(q^2)^5
= 1 * (32p^5) * 1 + 5 * (16p^4) * q^2 + 10 * (8p^3) * q^4 + 10 * (4p^2) * q^6 + 5 * (2p) * q^8 + 1 * 1 * q^10
= 32p^5 + 80p^4q^2 + 80p^3q^4 + 40p^2q^6 + 10pq^8 + q^10
c. (3p^2-q)^5:
n=5, x=3p^2, y=-q
(5 choose 0)(3p^2)^5(-q)^0 + (5 choose 1)(3p^2)^4(-q)^1 + (5 choose 2)(3p^2)^3(-q)^2 + (5 choose 3)(3p^2)^2(-q)^3 + (5 choose 4)(3p^2)^1(-q)^4 + (5 choose 5)(3p^2)^0(-q)^5
= 1 * (243p^10) * 1 + 5 * (81p^8) * (-q) + 10 * (27p^6) * q^2 + 10 * (9p^4) * (-q^3) + 5 * (3p^2) * q^4 + 1 * 1 * (-q^5)
= 243p^10 - 405p^8q + 270p^6q^2 - 90p^4q^3 + 15p^2q^4 - q^5