`f(x)=2x^(3)-5x^(2)-8x+20`
What are the zeros of function f defined by the above equation?

Answer

To find the zeros of the function \( f(x) = 2x^3 - 5x^2 - 8x + 20 \), we need to solve the equation \( f(x) = 0 \).

### Step 1: Set the function equal to zero
We start with the equation:
\[ 2x^3 - 5x^2 - 8x + 20 = 0 \]

### Step 2: Factor the polynomial
To factor the polynomial, we can look for common factors in the terms. We can group the terms as follows:
\[ f(x) = 2x^3 - 5x^2 - 8x + 20 \]
We can group the first two terms and the last two terms:
\[ = (2x^3 - 5x^2) + (-8x + 20) \]
Now, we can factor out \( x^2 \) from the first group and \(-4\) from the second group:
\[ = x^2(2x - 5) - 4(2x - 5) \]
Now, we can factor out the common term \( (2x - 5) \):
\[ = (2x - 5)(x^2 - 4) \]

### Step 3: Factor the quadratic expression
The expression \( x^2 - 4 \) is a difference of squares, which can be factored as:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = (2x - 5)(x - 2)(x + 2) \]

### Step 4: Set each factor to zero
To find the zeros, we set each factor equal to zero:
1. \( 2x - 5 = 0 \)
2. \( x - 2 = 0 \)
3. \( x + 2 = 0 \)

### Step 5: Solve each equation
1. For \( 2x - 5 = 0 \):
\[ 2x = 5 \implies x = \frac{5}{2} \]
2. For \( x - 2 = 0 \):
\[ x = 2 \]
3. For \( x + 2 = 0 \):
\[ x = -2 \]

### Conclusion
The zeros of the function \( f(x) \) are:
\[ x = \frac{5}{2}, \quad x = 2, \quad x = -2 \]