Rozwiązania w załącznikach
zad 29: [latex]w(x) = (ax + 2)(x^2 + bx - 2) - 2 cdot [(a - 2b)x^3 + 9x^2 - 6x - 2][/latex] [latex]w(x) = ax^3 + abx^2 - 2ax + 2x^2 + 2bx - 4 - 2(a - 2b)x^3 - 18x^2 + 12x + 4[/latex] [latex]w(x) = (-a + 4b)x^3 + (ab - 16)x^2 + (12 - 2a + 2b)x [/latex] [latex]4b - a = 0 qquad ab - 16 = 0 qquad 12 - 2a + 2b = 0[/latex] [latex]4b = a qquad 4b^2 - 16 = 0 qquad 12 - 8b + 2b = 0[/latex] [latex]a = 4b qquad b^2 = 4 qquad 6b = 12[/latex] [latex]b = 2 qquad a = 8[/latex] zad 32: wielomian jest podzielny przez dwumian x - 2 jeżeli [latex]w(2) = 0[/latex] [latex]3 cdot 2^3 + (a + b) cdot 2^2 - 27 cdot 2 + a - 4b = 0[/latex] [latex]24 + 4a + 4b - 54 + a - 4b = 0[/latex] [latex]5a - 30 = 0[/latex] [latex]a = 6[/latex] Ponadto [latex]w(1) = 24[/latex] [latex]3 cdot 1^3 + (a + b) cdot 1^2 - 27 cdot 1 + a - 4b = 24[/latex] [latex]3 + a + b - 27 + a - 4b = 24[/latex] [latex]2a - 3b - 24 = 24[/latex] [latex]b = -12[/latex] Zatem [latex]w(x) = 3x^3 - 6x^2 - 27x + 54 = 3(x^3 - 2x^2 - 9x + 18) = 3[x^2(x - 2) - 9(x - 2)] = 3(x - 2)(x^2 - 9) = 3(x - 2)(x - 3)(x + 3)[/latex] Pierwiastki: [latex]x = 2 qquad x = 3 qquad x = -3[/latex]
