%5E3.jpeg "(1+x^2)^3")
Exploring the Expansion of (1 + x^2)³
The expression (1 + x²)³ represents a polynomial raised to the power of three. This can be expanded using the binomial theorem, which provides a general formula for expanding expressions of the form (a + b)ⁿ.
Understanding the Binomial Theorem
The binomial theorem states:
(a + b)ⁿ = ∑(k=0 to n) [nCk * a^(n-k) * b^k]
where:
- nCk represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This represents the number of ways to choose k items from a set of n items.
Expanding (1 + x²)³
Applying the binomial theorem to our expression, we have:
(1 + x²)³ = ∑(k=0 to 3) [3Ck * 1^(3-k) * (x²) ^k]
Let's expand this term by term:
- k = 0: 3C0 * 1³ * (x²)⁰ = 1
- k = 1: 3C1 * 1² * (x²)¹ = 3x²
- k = 2: 3C2 * 1¹ * (x²)² = 3x⁴
- k = 3: 3C3 * 1⁰ * (x²)³ = x⁶
Adding these terms together, we obtain the expanded form:
(1 + x²)³ = 1 + 3x² + 3x⁴ + x⁶
Key Observations
- The expanded form is a polynomial of degree 6.
- The coefficients of the terms follow the pattern of Pascal's triangle: 1, 3, 3, 1.
- The exponents of 'x' increase by 2 in each subsequent term.
Applications
Understanding the expansion of (1 + x²)³ has various applications in mathematics, including:
- Calculus: This expansion can be used to find derivatives and integrals of the original expression.
- Algebra: It can be used to simplify more complex expressions involving (1 + x²)³.
- Polynomial functions: The expansion helps understand the behaviour of the function represented by (1 + x²)³.
By applying the binomial theorem and understanding its significance, we can effectively expand and analyze the expression (1 + x²)³. This knowledge can be further utilized in various mathematical contexts.