What is the difference between the values of the polynomial `7x-3x^2+7` at x=1 and x=2?

Options

A. -2
B. 2
C. 3
D. None of these

Answer

To find the difference between the values of the polynomial \(7x - 3x^2 + 7\) at \(x = 1\) and \(x = 2\), we can follow these steps:

### Step 1: Define the polynomial
Let \(f(x) = 7x - 3x^2 + 7\).

### Step 2: Calculate \(f(1)\)
Substituting \(x = 1\) into the polynomial:
\[ f(1) = 7(1) - 3(1^2) + 7 \]
Calculating each term:
\[ f(1) = 7 - 3(1) + 7 = 7 - 3 + 7 \]
Now, simplify:
\[ f(1) = 7 + 7 - 3 = 14 - 3 = 11 \]

### Step 3: Calculate \(f(2)\)
Now, substituting \(x = 2\) into the polynomial:
\[ f(2) = 7(2) - 3(2^2) + 7 \]
Calculating each term:
\[ f(2) = 14 - 3(4) + 7 = 14 - 12 + 7 \]
Now, simplify:
\[ f(2) = 14 - 12 + 7 = 2 + 7 = 9 \]

### Step 4: Find the difference \(f(1) - f(2)\)
Now we find the difference between the two values:
\[ \text{Difference} = f(1) - f(2) = 11 - 9 \]
Calculating the difference:
\[ \text{Difference} = 2 \]

### Final Answer
The difference between the values of the polynomial at \(x = 1\) and \(x = 2\) is \(2\).

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