The product of the polynomials `(x^(2)-x+2)` and `(x-1)` is __________.

Options

A. `x^(3)-2x^(2)+3x-2`
B. `x^(3)+3x^(2)-3x+2`
C. `x^(3)-2x+4x^(2)-6`
D. `x^(3)-2x^(2)+3x+2`

Answer

To find the product of the polynomials \( (x^2 - x + 2) \) and \( (x - 1) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s how to do it step by step:

### Step 1: Write the expression
We start with the expression:
\[ (x^2 - x + 2)(x - 1) \]

### Step 2: Distribute each term in the first polynomial
Now, we will distribute each term in the first polynomial \( (x^2 - x + 2) \) to each term in the second polynomial \( (x - 1) \).

1. **Multiply \( x^2 \) by \( x \)**:
\[ x^2 \cdot x = x^3 \]

2. **Multiply \( x^2 \) by \( -1 \)**:
\[ x^2 \cdot (-1) = -x^2 \]

3. **Multiply \( -x \) by \( x \)**:
\[ -x \cdot x = -x^2 \]

4. **Multiply \( -x \) by \( -1 \)**:
\[ -x \cdot (-1) = x \]

5. **Multiply \( 2 \) by \( x \)**:
\[ 2 \cdot x = 2x \]

6. **Multiply \( 2 \) by \( -1 \)**:
\[ 2 \cdot (-1) = -2 \]

### Step 3: Combine all the results
Now, we combine all the results from the multiplications:
\[ x^3 - x^2 - x^2 + x + 2x - 2 \]

### Step 4: Simplify the expression
Combine like terms:
- The \( x^2 \) terms: \( -x^2 - x^2 = -2x^2 \)
- The \( x \) terms: \( x + 2x = 3x \)

So, the expression simplifies to:
\[ x^3 - 2x^2 + 3x - 2 \]

### Final Answer
The product of the polynomials \( (x^2 - x + 2) \) and \( (x - 1) \) is:
\[ \boxed{x^3 - 2x^2 + 3x - 2} \]

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