Which one of the following is the second degree polynomial function f(x), where `f(0)=5 f(-1)=10 and f(1)=6=?`

Options

A. `5x^(2)-2x+5`
B. `3x^(2)-2x-5`
C. `3x^(2)-2x+5`
D. `3x^(2)-10x+5`

Answer

To find the second degree polynomial function \( f(x) \) that satisfies the conditions \( f(0) = 5 \), \( f(-1) = 10 \), and \( f(1) = 6 \), we can start by expressing the polynomial in its general form:

### Step 1: Write the general form of a second degree polynomial
A second degree polynomial can be expressed as:
\[ f(x) = ax^2 + bx + c \]

### Step 2: Use the condition \( f(0) = 5 \)
Substituting \( x = 0 \) into the polynomial gives:
\[ f(0) = a(0)^2 + b(0) + c = c \]
Since \( f(0) = 5 \), we have:
\[ c = 5 \]

### Step 3: Substitute \( c \) back into the polynomial
Now we can rewrite the polynomial as:
\[ f(x) = ax^2 + bx + 5 \]

### Step 4: Use the condition \( f(-1) = 10 \)
Substituting \( x = -1 \) into the polynomial gives:
\[ f(-1) = a(-1)^2 + b(-1) + 5 = a - b + 5 \]
Setting this equal to 10:
\[ a - b + 5 = 10 \]
This simplifies to:
\[ a - b = 5 \quad \text{(Equation 1)} \]

### Step 5: Use the condition \( f(1) = 6 \)
Substituting \( x = 1 \) into the polynomial gives:
\[ f(1) = a(1)^2 + b(1) + 5 = a + b + 5 \]
Setting this equal to 6:
\[ a + b + 5 = 6 \]
This simplifies to:
\[ a + b = 1 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations
Now we have a system of two equations:
1. \( a - b = 5 \)
2. \( a + b = 1 \)

We can add these two equations:
\[ (a - b) + (a + b) = 5 + 1 \]
This simplifies to:
\[ 2a = 6 \implies a = 3 \]

Now substitute \( a = 3 \) back into Equation 2:
\[ 3 + b = 1 \implies b = 1 - 3 = -2 \]

### Step 7: Write the final polynomial
Now we have \( a = 3 \), \( b = -2 \), and \( c = 5 \). Thus, the polynomial is:
\[ f(x) = 3x^2 - 2x + 5 \]

### Conclusion
The second degree polynomial function \( f(x) \) that satisfies the given conditions is:
\[ f(x) = 3x^2 - 2x + 5 \]