%5E3-binomial-expansion.jpeg "(1+x)^3 Binomial Expansion")
Understanding the Binomial Expansion of (1+x)³
The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)ⁿ. This article focuses on the specific case of (1 + x)³, showcasing the steps and key concepts involved in its expansion.
The Binomial Theorem
The binomial theorem states that for any real numbers a and b, and any non-negative integer n, the following holds:
(a + b)ⁿ = ∑ (n choose k) * a^(n-k) * b^k
where the summation is taken over all values of k from 0 to n, and (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Expanding (1 + x)³
Applying the binomial theorem to (1 + x)³, we have:
(1 + x)³ = ∑ (3 choose k) * 1^(3-k) * x^k
Let's calculate each term of the expansion:
- k = 0: (3 choose 0) * 1³ * x⁰ = 1 * 1 * 1 = 1
- k = 1: (3 choose 1) * 1² * x¹ = 3 * 1 * x = 3x
- k = 2: (3 choose 2) * 1¹ * x² = 3 * 1 * x² = 3x²
- k = 3: (3 choose 3) * 1⁰ * x³ = 1 * 1 * x³ = x³
Therefore, the expanded form of (1 + x)³ is:
(1 + x)³ = 1 + 3x + 3x² + x³
Key Observations
- Coefficients: The coefficients in the expansion (1, 3, 3, 1) correspond to the fourth row of Pascal's triangle. This pattern holds true for any binomial expansion.
- Exponents: The exponents of x increase from 0 to n (in this case, 3).
- Symmetry: The coefficients are symmetrical. Notice that the first and last coefficients are equal, as are the second and second-to-last coefficients.
Applications
The binomial expansion of (1 + x)³ has various applications in different fields:
- Calculus: It helps in deriving derivatives and integrals of functions involving (1 + x)³.
- Statistics: It plays a role in probability calculations involving binomial distributions.
- Financial mathematics: It's used to model compound interest and other financial growth scenarios.
By understanding the binomial theorem and its application to (1 + x)³, you gain valuable tools for solving problems in various mathematical and scientific domains.