%5E2.jpeg "(x^2+4)^2")
Exploring the Expression (x^2 + 4)^2
The expression (x^2 + 4)^2 represents the square of the binomial (x^2 + 4). Understanding how to expand and simplify such expressions is crucial in algebra and calculus.
Expanding the Expression
To expand (x^2 + 4)^2, we can apply the FOIL method (First, Outer, Inner, Last) or use the square of a binomial formula:
(a + b)^2 = a^2 + 2ab + b^2
Using FOIL:
- First: x^2 * x^2 = x^4
- Outer: x^2 * 4 = 4x^2
- Inner: 4 * x^2 = 4x^2
- Last: 4 * 4 = 16
Combining these terms: x^4 + 4x^2 + 4x^2 + 16
Using the Square of a Binomial Formula:
- a = x^2
- b = 4
(x^2 + 4)^2 = (x^2)^2 + 2(x^2)(4) + (4)^2 = x^4 + 8x^2 + 16
Therefore, the expanded form of (x^2 + 4)^2 is x^4 + 8x^2 + 16.
Applications
The expression (x^2 + 4)^2 has various applications in different mathematical fields:
- Algebra: Simplifying equations and solving for unknown variables.
- Calculus: Finding derivatives and integrals of functions involving (x^2 + 4)^2.
- Geometry: Describing the area of certain geometric shapes.
- Physics: Modeling physical phenomena, such as the motion of objects.
Key Points
- (x^2 + 4)^2 is a perfect square trinomial.
- It can be expanded using the FOIL method or the square of a binomial formula.
- The expanded form is x^4 + 8x^2 + 16.
Understanding the expansion and simplification of (x^2 + 4)^2 is an essential foundation for further algebraic and calculus studies.