Identities 2

1. (ax + b)(x + c) = 3x^2 + 5x + 2
ax^2 + acx + bx + bc = 3x^2 + 5x + 2
By comparing coefficents,
a = 3
ac + b = 5 => 3c + b = 5 => b = 5 - 3c ... (1)
bc = 2 ... (2)
Use (1), (5 - 3c)c = 2
-3c^2 + 5c - 2 = 0
3c^2 - 5c + 2 = 0
(c - 1)(3c - 2) = 0
c = 1 or c = 2/3
when c = 1, b = 2 => (ax + b)(x + c) = (3x + 2)(x + 1)
when c = 2/3, b = 3 => (ax + b)(x + c) = (3x + 3)(x + 2/3) = (3x + 2)(x + 1)
2. ax^3 - x^2 - 4x + 5 = (2x - 1)(bx^2 + cx - 5)
(2x - 1)(bx^2 + cx - 5)
= 2bx^3 + 2cx^2 - 10x - bx^2 - cx + 5
= 2bx^3 + (2c - b)x^2 - (10 + c)x + 5
By comparing coefficients,
a = 2b ... (1)
-1 = 2c - b ... (2)
-4 = -10 - c ... (3)
(3) => c = -6
(2) => -1 = -12 - b => b = -11
(1) => a = -22
3. a(x - 1)(x - 2) + b(x - 2)(x - 3) + c(x - 3)(x - 1) = x
Put x = 1, b(-1)(-2) = 1 => b = 1/2
Put x = 2, c(-1)(1) = 2 => c = -2
Put x = 3, a(2)(1) = 3 => a = 3/2