This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Ili kutatua swali hili, tutatumia Theorem ya Mabaki (Remainder Theorem). Step 1: Tumia masharti ya kugawanyika. Polynomial P(x) = ax^3 - bx^2 - cx + 3 inagawanyika na x^2 - 1. Hii inamaanisha inagawanyika na (x-1) na (x+1). Kwa hiyo, P(1) = 0 na P(-1) = 0. Pia, 2x - 3 ni kigezo, hivyo P((3)/(2)) = 0. Step 2: Unda milinganyo kutoka kwa masharti. Kutoka P(1) = 0: a(1)^3 - b(1)^2 - c(1) + 3 = 0 a - b - c + 3 = 0 (Mlinganyo 1) Kutoka P(-1) = 0: a(-1)^3 - b(-1)^2 - c(-1) + 3 = 0 -a - b + c + 3 = 0 (Mlinganyo 2) Kutoka P((3)/(2)) = 0: a((3)/(2))^3 - b((3)/(2))^2 - c((3)/(2)) + 3 = 0 a((27)/(8)) - b((9)/(4)) - c((3)/(2)) + 3 = 0 Kuzidisha mlinganyo mzima kwa 8 ili kuondoa sehemu: 27a - 18b - 12c + 24 = 0 Kugawanya mlinganyo mzima kwa 3: 9a - 6b - 4c + 8 = 0 (Mlinganyo 3) Step 3: Tatua mfumo wa milinganyo. Tutaongeza Mlinganyo 1 na Mlinganyo 2: (a - b - c) + (-a - b + c) = -3 + (-3) -2b = -6 b = 3 Badilisha b=3 katika Mlinganyo 1: a - 3 - c + 3 = 0 a - c = 0 a = c Badilisha b=3 na a=c katika Mlinganyo 3: 9a - 6(3) - 4a + 8 = 0 9a - 18 - 4a + 8 = 0 5a - 10 = 0 5a = 10 a = 2 Kwa kuwa a=c, basi c=2. Kwa hiyo, maadili ni a=2, b=3, na c=2. Maadili ya a, b, na c ni: a = 2 b = 3 c = 2 Tuma swali linalofuata 📸