Write `(a ^(2) - b ^(2))/( a ^(2) - 3 ab + 2b ^(2))` in the simplest form

Options

A. `((a + b) ^(2))/((a - 2b))`
B. `(a +b)/( a - 2b)`
C. `((a - b))/( (a - 2b))`
D. `((a + b ))/( (a + 2b ))`

Answer

To simplify the expression \((a^2 - b^2)/(a^2 - 3ab + 2b^2)\), we can follow these steps:

### Step 1: Factor the Numerator
The numerator is \(a^2 - b^2\). We can use the difference of squares formula, which states that \(x^2 - y^2 = (x - y)(x + y)\).

So, we can factor the numerator as:
\[ a^2 - b^2 = (a - b)(a + b) \]

### Step 2: Factor the Denominator
The denominator is \(a^2 - 3ab + 2b^2\). We need to factor this quadratic expression. We look for two numbers that multiply to \(2b^2\) (the product of the last term) and add to \(-3ab\) (the coefficient of the middle term).

The numbers \(-2b\) and \(-b\) satisfy these conditions:
- \(-2b \times -b = 2b^2\)
- \(-2b + -b = -3b\)

Thus, we can factor the denominator as:
\[ a^2 - 3ab + 2b^2 = (a - 2b)(a - b) \]

### Step 3: Rewrite the Expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{(a - b)(a + b)}{(a - 2b)(a - b)} \]

### Step 4: Cancel Common Factors
We notice that \((a - b)\) is a common factor in both the numerator and the denominator, so we can cancel it out:
\[ \frac{(a + b)}{(a - 2b)} \]

### Final Simplified Form
Thus, the expression in its simplest form is:
\[ \frac{a + b}{a - 2b} \]

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