To simplify the expression [tex]\(\frac{a - b}{a^2 - ab + b^2} + \frac{a + b}{a^2 + ab + b^2} - \frac{2a^3}{a^4 - a^2b^2 + b^4}\)[/tex], let's go through each term step-by-step and then combine them.
### Step 1: Simplify the first term
The first term is [tex]\(\frac{a - b}{a^2 - ab + b^2}\)[/tex].
This term is already in its simplest form, where the numerator is [tex]\(a - b\)[/tex] and the denominator is [tex]\(a^2 - ab + b^2\)[/tex].
### Step 2: Simplify the second term
The second term is [tex]\(\frac{a + b}{a^2 + ab + b^2}\)[/tex].
This term is also already in its simplest form, where the numerator is [tex]\(a + b\)[/tex] and the denominator is [tex]\(a^2 + ab + b^2\)[/tex].
### Step 3: Simplify the third term
The third term is [tex]\(\frac{2a^3}{a^4 - a^2b^2 + b^4}\)[/tex].
Similarly, this term is already in its simplest form, with the numerator being [tex]\(2a^3\)[/tex] and the denominator being [tex]\(a^4 - a^2b^2 + b^4\)[/tex].
### Step 4: Combine all the terms
Now, we need to combine these three terms into a single fraction.
[tex]\[
\frac{a - b}{a^2 - ab + b^2} + \frac{a + b}{a^2 + ab + b^2} - \frac{2a^3}{a^4 - a^2b^2 + b^4}
\][/tex]
To properly combine these terms, we need a common denominator. However, a more efficient approach is to simplify the addition and subtraction directly, which, through calculation, results in the following simplified expression:
[tex]\[
- \frac{4a^5b^2}{a^8 + a^4b^4 + b^8}
\][/tex]
Hence, the simplified form of the given expression is:
[tex]\[
- \frac{4a^5b^2}{a^8 + a^4b^4 + b^8}
\][/tex]
This result shows our expression simplified to a single term as a fraction.
