(a-2)(a%5E2%2B2a%2B4)(a%5E2-2a%2B4).jpeg "(a+2)(a-2)(a^2+2a+4)(a^2-2a+4)")
Simplifying the Expression (a+2)(a-2)(a^2+2a+4)(a^2-2a+4)
This expression involves a pattern that allows us to simplify it significantly. Let's break it down:
Recognizing the Pattern
The expression consists of four factors:
- (a+2)(a-2): This is a difference of squares pattern (a² - b²).
- (a² + 2a + 4): This is a sum of cubes pattern (a³ + b³).
- (a² - 2a + 4): This is a difference of cubes pattern (a³ - b³).
Applying the Patterns
Let's apply the algebraic patterns to simplify:
-
Difference of Squares: (a + 2)(a - 2) = a² - 2² = a² - 4
-
Sum of Cubes: (a² + 2a + 4) = (a + 2)(a² - 2a + 4)
-
Difference of Cubes: (a² - 2a + 4) = (a - 2)(a² + 2a + 4)
Combining the Results
Now we can substitute the simplified expressions back into the original one:
(a+2)(a-2)(a^2+2a+4)(a^2-2a+4) = (a² - 4) * (a + 2)(a² - 2a + 4) * (a - 2)(a² + 2a + 4)
Notice that the factors (a + 2) and (a - 2) appear twice. Let's group them:
= (a² - 4) * [(a + 2)(a - 2)] * [(a² - 2a + 4)(a² + 2a + 4)]
= (a² - 4) * (a² - 4) * (a⁴ + 4a² + 16)
Final Simplified Expression
Finally, we can multiply the remaining factors:
= (a⁴ - 8a² + 16) * (a⁴ + 4a² + 16)
This is the simplified form of the original expression. While it might seem complex, it's significantly easier to work with than the initial form, especially when performing further operations.