%5E4.jpeg "(x^2+1)^4")
Expanding (x^2 + 1)^4
The expression (x² + 1)⁴ represents the fourth power of the binomial (x² + 1). To expand this, we can use the Binomial Theorem:
The Binomial Theorem:
For any real numbers a and b, and any non-negative integer n:
(a + b)ⁿ = ∑(k=0 to n) [nCk * a^(n-k) * b^k]
where nCk is the binomial coefficient, calculated as n!/(k!(n-k)!).
Applying the Binomial Theorem to (x² + 1)⁴:
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Identify a and b: In our case, a = x² and b = 1.
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Calculate the binomial coefficients: We need to calculate the coefficients for n = 4:
- ⁴C₀ = 1
- ⁴C₁ = 4
- ⁴C₂ = 6
- ⁴C₃ = 4
- ⁴C₄ = 1
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Substitute the values into the theorem:
(x² + 1)⁴ = 1 * (x²)⁴ * (1)⁰ + 4 * (x²)³ * (1)¹ + 6 * (x²)² * (1)² + 4 * (x²)¹ * (1)³ + 1 * (x²)⁰ * (1)⁴
- Simplify the expression:
(x² + 1)⁴ = x⁸ + 4x⁶ + 6x⁴ + 4x² + 1
Therefore, the expanded form of (x² + 1)⁴ is x⁸ + 4x⁶ + 6x⁴ + 4x² + 1.
Understanding the result:
This expanded form represents a polynomial of degree 8. It has five terms, each with a coefficient determined by the binomial theorem. This polynomial represents the product of four identical binomials: (x² + 1) * (x² + 1) * (x² + 1) * (x² + 1).
Applications:
Understanding how to expand binomials like (x² + 1)⁴ is important in many areas of mathematics, including:
- Algebra: Simplifying expressions and solving equations
- Calculus: Finding derivatives and integrals
- Statistics: Modeling and analyzing data
The Binomial Theorem provides a powerful tool for handling these types of expressions and exploring their mathematical properties.