((2x-1)(x-1)^(4)(x-2)^(4))/((x-2)(x-4)^(4)) =0

Jasonbradley

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Jun 16, 2024

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Solving the Equation: ((2x-1)(x-1)^(4)(x-2)^(4))/((x-2)(x-4)^(4)) = 0

This equation involves rational expressions and presents a straightforward solution process. Here's how to solve it:

Understanding the Equation:

The equation represents a fraction where the numerator and denominator are polynomials. The equation is true when the numerator equals zero.

Solving for x:

1. Set the numerator equal to zero: (2x-1)(x-1)^(4)(x-2)^(4) = 0

2. Apply the Zero Product Property: The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

   • 2x - 1 = 0
   • (x - 1)^(4) = 0
   • (x - 2)^(4) = 0

3. Solve for x in each equation:

   • 2x - 1 = 0 => x = 1/2
   • (x - 1)^(4) = 0 => x = 1
   • (x - 2)^(4) = 0 => x = 2

Solutions:

Therefore, the solutions to the equation ((2x-1)(x-1)^(4)(x-2)^(4))/((x-2)(x-4)^(4)) = 0 are x = 1/2, x = 1, and x = 2.

Important Note:

While x = 2 would make the numerator zero, it also makes the denominator zero. This would result in an undefined expression, meaning it is not a valid solution to the original equation.

Therefore, the final solutions are x = 1/2 and x = 1.