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((1-x)^(3)(x+2)^(4))/((x+9)^(2)(x-8)) =0

3 min read Jun 16, 2024
((1-x)^(3)(x+2)^(4))/((x+9)^(2)(x-8)) =0

Solving the Equation: ((1-x)^(3)(x+2)^(4))/((x+9)^(2)(x-8)) = 0

This equation presents a rational expression set equal to zero. To find the solutions, we need to understand the properties of fractions and how they relate to zero.

Key Concepts:

  • A fraction equals zero only if the numerator is zero. This is because any number divided by a non-zero number will result in a non-zero value.
  • We cannot have a zero in the denominator. A fraction with a zero in the denominator is undefined.

Solving the Equation:

  1. Focus on the numerator: Since the fraction is equal to zero, the numerator must be zero. Therefore, we need to solve the equation: (1-x)^(3)(x+2)^(4) = 0

  2. 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. Applying this to our equation, we have:

    • (1-x)^(3) = 0 or
    • (x+2)^(4) = 0

  3. Solve for x:

    • (1-x)^(3) = 0

      • Taking the cube root of both sides: 1-x = 0
      • Solving for x: x = 1

    • (x+2)^(4) = 0

      • Taking the fourth root of both sides: x+2 = 0
      • Solving for x: x = -2

Important Note: We don't need to consider the denominator (x+9)^(2)(x-8) for finding the solutions because the fraction is zero only if the numerator is zero.

Solutions: The solutions to the equation ((1-x)^(3)(x+2)^(4))/((x+9)^(2)(x-8)) = 0 are:

  • x = 1
  • x = -2

Understanding the Solutions: These solutions represent the x-values where the function represented by the expression equals zero. At these points, the graph of the function would intersect the x-axis.

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