+349 In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of a variable t whose roots are t = a, t = b, and t = c.
+10 Identify the variables in terms of a, b, and c:
Given a = x/2, b = 5y, and c = -4z:
x = 2a, y = b/5, z = -c/4.
Substitute x, y, z:
3(2a) + 4(b/5) + 30(-c/4) = -60 → 120a + 16b - 150c = -1200 → 120a + 16b - 150c + 1200 = 0.
2(2a)(b/5) + 42(2a)(-c/4) - 16(b/5)(-c/4) = 68 → 16ab - 210ac + 16bc = 1360.
5(2a)(b/5)(-c/4) = 56 → abc = -112.
Form the polynomial:
Roots: a, b, c → (t - a)(t - b)(t - c) = 0.
Expanding: t^3 - (a + b + c)t^2 + (ab + ac + bc)t - abc = 0.
Coefficients:
Let p = a + b + c, q = ab + ac + bc, r = abc. Polynomial: t^3 - pt^2 + qt - r = 0.
Final form: t^3 - pt^2 + qt + 112 = 0.
