For all non-negative x, let f(x)=`x^(3)-8` and g(x)=x-2. For how many integer values of x is f(x)=g(x)? NC

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem of finding how many integer values of \( x \) satisfy the equation \( f(x) = g(x) \) where \( f(x) = x^3 - 8 \) and \( g(x) = x - 2 \), we will follow these steps:

### Step 1: Set the functions equal to each other
We start with the equation:
\[ f(x) = g(x) \]
Substituting the definitions of \( f(x) \) and \( g(x) \):
\[ x^3 - 8 = x - 2 \]

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ x^3 - x - 8 + 2 = 0 \]
This simplifies to:
\[ x^3 - x - 6 = 0 \]

### Step 3: Factor the polynomial
To find the integer solutions, we can use the Rational Root Theorem or test for possible integer roots. Testing \( x = 2 \):
\[ f(2) = 2^3 - 2 - 6 = 8 - 2 - 6 = 0 \]
Thus, \( x = 2 \) is a root.

### Step 4: Factor out \( (x - 2) \)
Now we can factor \( x^3 - x - 6 \) using synthetic division or polynomial long division by \( (x - 2) \):
\[ x^3 - x - 6 = (x - 2)(x^2 + 2x + 3) \]

### Step 5: Solve the quadratic equation
Next, we need to solve the quadratic equation:
\[ x^2 + 2x + 3 = 0 \]
Using the discriminant \( D = b^2 - 4ac \):
\[ D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
Since the discriminant is negative, there are no real solutions from this quadratic equation.

### Step 6: Conclusion
The only integer solution we found is from \( x - 2 = 0 \), which gives:
\[ x = 2 \]
Thus, there is only **one integer value** of \( x \) that satisfies \( f(x) = g(x) \).

### Final Answer
The number of integer values of \( x \) for which \( f(x) = g(x) \) is:
\[ \boxed{1} \]