The given equation is: (x - b)(x - c)(x - 3)(x - 1)(x - 1)(x - b) = 0


To find the roots of the equation, we need to solve it step by step.


We can simplify the given equation by expanding it using the distributive property.

(x - b)(x - c)(x - 3)(x - 1)(x - 1)(x - b) = 0
Expanding the equation:
(x² - cx - bx + bc)(x - 3)(x - 1)(x - 1)(x - b) = 0
(x² - (c + b)x + bc)(x - 3)(x - 1)(x - 1)(x - b) = 0
(x² - (c + b)x + bc)(x - 3)(x² - 2x + 1)(x - b) = 0
(x² - (c + b)x + bc)(x³ - 2x² + x - 3x² + 6x - 3)(x - b) = 0
(x² - (c + b)x + bc)(x³ - 5x² + 7x - 3)(x - b) = 0
(x⁵ - 5x⁴ + 7x³ - 3x² - (c + b)x⁴ + 5(c + b)x³ - 7(c + b)x² + 3(c + b)x)(x - b) = 0
(x⁵ - (5 + c + b)x⁴ + (7 - 5(c + b))x³ - (3 + 7(c + b))x² + 3(c + b)x)(x - b) = 0


According to the zero product property, if a product of factors is equal to zero, then at least one of the factors must be equal to zero. Therefore, we can set each factor equal to zero and solve for x.

Setting each factor equal to zero:
1. x⁵ - (5 + c + b)x⁴ + (7 - 5(c + b))x³ - (3 + 7(c + b))x² + 3(c + b)x = 0
2. x - b = 0


1. x⁵ - (5 + c + b)x⁴ + (7 - 5(c + b))x³ - (3 + 7(c + b))x² + 3(c + b)x = 0
This equation is a fifth-degree polynomial, and finding its roots analytically can be complex. It may require numerical methods or calculators to find the roots.

2. x - b = 0
Solving this equation, we get:
x = b

Roots are 1,1,1