Questions: Find all real zeros of the function. h(x)=-4 x(x-2)(x^2-25)

Solution Steps

The function is given as: \[ h(x) = -4x(x-2)(x^{2}-25) \] The factors of the function are \( -4x \), \( (x-2) \), and \( (x^{2}-25) \).

To find the zeros of the function, set each factor equal to zero:

1. \( -4x = 0 \)
2. \( x-2 = 0 \)
3. \( x^{2}-25 = 0 \)

1. Solve \( -4x = 0 \): \[ -4x = 0 \implies x = 0 \]
2. Solve \( x-2 = 0 \): \[ x-2 = 0 \implies x = 2 \]
3. Solve \( x^{2}-25 = 0 \): \[ x^{2}-25 = 0 \implies x^{2} = 25 \implies x = \pm 5 \]

The real zeros of the function are: \[ x = 0, \, x = 2, \, x = 5, \, x = -5 \]

zero(s): \[ \boxed{0, 2, 5, -5} \]

Final Answer