Suppose `a in R`. Define f and g as follows:
`f(x) =(a^(2) - 4a + 3)x^(2) + (a^(2) -1)a AA x in R` and `g(x) = (a^(2) - 5a + 6)x^(3) + (a^(2) -3a+ 2)x + (a^(3) -1) AA x in R`
The number of values of a for which `f(x) = g(x) AA x in R`

Options

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

Answer

To solve the problem, we need to find the number of values of \( a \) for which the functions \( f(x) \) and \( g(x) \) are equal for all \( x \in \mathbb{R} \).

### Step 1: Write down the functions
The functions are defined as:
\[ f(x) = (a^2 - 4a + 3)x^2 + (a^2 - 1)a \]
\[ g(x) = (a^2 - 5a + 6)x^3 + (a^2 - 3a + 2)x + (a^3 - 1) \]

### Step 2: Set the functions equal to each other
We need to set \( f(x) = g(x) \):
\[ (a^2 - 4a + 3)x^2 + (a^2 - 1)a = (a^2 - 5a + 6)x^3 + (a^2 - 3a + 2)x + (a^3 - 1) \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ 0 = (a^2 - 5a + 6)x^3 + (a^2 - 3a + 2)x + (a^3 - 1) - (a^2 - 4a + 3)x^2 - (a^2 - 1)a \]

### Step 4: Group terms by powers of \( x \)
Now we will group the terms by the powers of \( x \):
- Coefficient of \( x^3 \): \( a^2 - 5a + 6 \)
- Coefficient of \( x^2 \): \( -(a^2 - 4a + 3) \)
- Coefficient of \( x \): \( a^2 - 3a + 2 \)
- Constant term: \( a^3 - 1 - (a^2 - 1)a \)

### Step 5: Set coefficients equal to zero
For \( f(x) \) and \( g(x) \) to be equal for all \( x \), each coefficient must be equal to zero:
1. \( a^2 - 5a + 6 = 0 \)
2. \( -(a^2 - 4a + 3) = 0 \)
3. \( a^2 - 3a + 2 = 0 \)
4. \( a^3 - 1 - (a^2 - 1)a = 0 \)

### Step 6: Solve each equation
1. **From \( a^2 - 5a + 6 = 0 \)**:
\[ (a - 2)(a - 3) = 0 \implies a = 2 \text{ or } a = 3 \]

2. **From \( -(a^2 - 4a + 3) = 0 \)**:
\[ a^2 - 4a + 3 = 0 \implies (a - 1)(a - 3) = 0 \implies a = 1 \text{ or } a = 3 \]

3. **From \( a^2 - 3a + 2 = 0 \)**:
\[ (a - 1)(a - 2) = 0 \implies a = 1 \text{ or } a = 2 \]

4. **From \( a^3 - 1 - (a^2 - 1)a = 0 \)**:
\[ a^3 - 1 - a^3 + a = 0 \implies a - 1 = 0 \implies a = 1 \]

### Step 7: Combine solutions
The possible values of \( a \) from all equations are:
- From the first equation: \( 2, 3 \)
- From the second equation: \( 1, 3 \)
- From the third equation: \( 1, 2 \)
- From the fourth equation: \( 1 \)

The unique values of \( a \) that satisfy all equations are \( 1, 2, 3 \).

### Conclusion
Thus, the number of values of \( a \) for which \( f(x) = g(x) \) for all \( x \in \mathbb{R} \) is **3**.

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