(2x%5E2%2B5).jpeg "0 5x(2x^2-5)(2x^2+5)")
Factoring and Simplifying the Expression: 0.5x(2x^2-5)(2x^2+5)
This expression involves a combination of multiplication and factoring. Let's break it down step by step to understand how to simplify it.
Understanding the Structure
The expression is a product of three factors:
- 0.5x: This is a simple linear term.
- (2x^2 - 5): This is a difference of squares pattern.
- (2x^2 + 5): This is also a difference of squares pattern.
Applying the Difference of Squares Pattern
The difference of squares pattern states that: a^2 - b^2 = (a + b)(a - b)
Using this pattern, we can factor both (2x^2 - 5) and (2x^2 + 5):
- (2x^2 - 5) = (√2x + √5)(√2x - √5)
- (2x^2 + 5) = (√2x + √5)(√2x - √5)
Simplifying the Expression
Now, let's substitute these factored forms back into the original expression:
0.5x(2x^2 - 5)(2x^2 + 5) = 0.5x (√2x + √5)(√2x - √5) (√2x + √5)(√2x - √5)
Notice that we have two identical pairs of factors: (√2x + √5) and (√2x - √5). This allows us to simplify further:
0.5x (√2x + √5)(√2x - √5) (√2x + √5)(√2x - √5) = 0.5x [(√2x + √5)(√2x - √5)]^2
Applying the difference of squares pattern again:
0.5x [(√2x + √5)(√2x - √5)]^2 = 0.5x [(2x^2 - 5)]^2
Expanding and Final Result
Finally, we can expand the square:
0.5x [(2x^2 - 5)]^2 = 0.5x (4x^4 - 20x^2 + 25)
This is the simplified form of the original expression.
Therefore, 0.5x(2x^2-5)(2x^2+5) is equivalent to 0.5x(4x^4 - 20x^2 + 25).