The coefficient of `x^10` in `(1-2x + 3x^2)/(1-x)` is

Options

A. 1
B. 2
C. 3
D. `-2`

Answer

To find the coefficient of \( x^{10} \) in the expression \( \frac{1 - 2x + 3x^2}{1 - x} \), we will follow these steps:

### Step 1: Rewrite the Expression
We can rewrite the expression as:
\[ (1 - 2x + 3x^2)(1 - x)^{-1} \]
This is because \( \frac{1}{1 - x} \) can be expressed as the power series \( 1 + x + x^2 + x^3 + \ldots \).

### Step 2: Expand \( (1 - x)^{-1} \)
Using the formula for the geometric series, we have:
\[ (1 - x)^{-1} = 1 + x + x^2 + x^3 + x^4 + x^5 + \ldots \]
We will consider terms up to \( x^{10} \) since we are interested in the coefficient of \( x^{10} \).

### Step 3: Multiply the Two Expressions
Now we need to multiply \( 1 - 2x + 3x^2 \) with the series we expanded:
\[ (1 - 2x + 3x^2)(1 + x + x^2 + x^3 + \ldots) \]

### Step 4: Identify Terms Contributing to \( x^{10} \)
We will find the contributions to \( x^{10} \) from each term in \( 1 - 2x + 3x^2 \):
- From \( 1 \): The term \( 1 \cdot x^{10} \) contributes \( 1 \).
- From \( -2x \): The term \( -2x \cdot x^9 \) contributes \( -2 \).
- From \( 3x^2 \): The term \( 3x^2 \cdot x^8 \) contributes \( 3 \).

### Step 5: Combine the Contributions
Now we combine the contributions:
\[ 1 - 2 + 3 = 2 \]

### Conclusion
Thus, the coefficient of \( x^{10} \) in the expression \( \frac{1 - 2x + 3x^2}{1 - x} \) is \( 2 \).

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