(x^2+1)^4

3 min read Jun 17, 2024
(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:

  1. Identify a and b: In our case, a = x² and b = 1.

  2. Calculate the binomial coefficients: We need to calculate the coefficients for n = 4:

    • ⁴C₀ = 1
    • ⁴C₁ = 4
    • ⁴C₂ = 6
    • ⁴C₃ = 4
    • ⁴C₄ = 1
  3. 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

  1. 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.

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