%5E2.jpeg "(2x^2+3)^2")
Expanding the Square: (2x^2 + 3)^2
The expression (2x^2 + 3)^2 represents the square of a binomial. To expand this expression, we can use the following methods:
1. The FOIL Method
The FOIL method stands for First, Outer, Inner, Last. This method helps us multiply two binomials.
Let's apply this to our expression:
- First: (2x^2) * (2x^2) = 4x^4
- Outer: (2x^2) * (3) = 6x^2
- Inner: (3) * (2x^2) = 6x^2
- Last: (3) * (3) = 9
Now, we add all the terms together: 4x^4 + 6x^2 + 6x^2 + 9
Combining like terms, we get:
(2x^2 + 3)^2 = 4x^4 + 12x^2 + 9
2. The Binomial Theorem
The Binomial Theorem provides a general formula for expanding any binomial raised to a power:
(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + b^n
Where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to our expression, we have:
(2x^2 + 3)^2 = (2x^2)^2 + 2(2x^2)(3) + 3^2
Simplifying, we get:
(2x^2 + 3)^2 = 4x^4 + 12x^2 + 9
Conclusion
Both the FOIL method and the Binomial Theorem provide us with the same expanded form of (2x^2 + 3)^2, which is 4x^4 + 12x^2 + 9. Choosing the method depends on your preference and the complexity of the expression you're working with.