(x2-2x-8).(x-1)2.(x+5)6 0

3 min read Jun 17, 2024

Solving the Polynomial Equation: (x²-2x-8)(x-1)²(x+5)⁶ = 0

This equation involves a product of multiple factors set equal to zero. To solve for the values of x that satisfy the equation, we can use the Zero Product Property: If the product of multiple factors is zero, at least one of the factors must be zero.

Let's break down each factor and find their roots:

Factor 1: (x²-2x-8)

This is a quadratic expression. We can factor it as:

(x²-2x-8) = (x-4)(x+2)

Therefore, this factor equals zero when:

x - 4 = 0 => x = 4
x + 2 = 0 => x = -2

Factor 2: (x-1)²

This factor is a perfect square trinomial. It equals zero when:

x - 1 = 0 => x = 1

Since the factor is squared, this solution has a multiplicity of 2.

Factor 3: (x+5)⁶

This factor is also a power of a linear expression. It equals zero when:

x + 5 = 0 => x = -5

This solution has a multiplicity of 6.

Solutions to the Equation

Combining all the solutions from each factor, the complete set of solutions for the equation (x²-2x-8)(x-1)²(x+5)⁶ = 0 is:

x = 4, x = -2, x = 1, x = -5

It's important to remember that the solutions x = 1 and x = -5 are repeated solutions due to their multiplicities.

This means that the equation has a total of 9 solutions when considering the multiplicities:

4 distinct solutions (x = 4, x = -2, x = 1, x = -5)
2 solutions for x = 1
6 solutions for x = -5