Binomial Expansion Calculator

What is the binomial theorem?

The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ where n is a non-negative integer. It states that (a + b)ⁿ = Σ_k=0ⁿ C(n,k) a^n-k b^k.

How do you expand (a+b)^n?

Step 1: Determine the binomial coefficients using Pascal's triangle or the formula C(n,k) = n! / (k! × (n-k)!).

Step 2: Write each term as C(n,k) × a^n-k × b^k.

Step 3: Sum all terms from k=0 to n.

What are binomial coefficients?

Binomial coefficients are the numbers that appear in binomial expansions. They are denoted as C(n,k) or (n choose k). They can be found using Pascal's triangle or the formula C(n,k) = n! / (k! × (n-k)!).

How do you expand (x+1)^4?

Using the binomial theorem:

(x + 1)⁴ = C(4,0)x⁴ + C(4,1)x³ + C(4,2)x² + C(4,3)x + C(4,4)

= 1x⁴ + 4x³ + 6x² + 4x + 1

What is Pascal's triangle?

Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The nth row gives the coefficients for the expansion of (a + b)ⁿ. Row 0: [1], Row 1: [1,1], Row 2: [1,2,1], Row 3: [1,3,3,1], etc.

How do you expand binomials with negative exponents?

The binomial theorem extends to negative and fractional exponents using the generalized binomial theorem. For (a + b)ⁿ with n < 0, the expansion becomes an infinite series. This calculator focuses on non-negative integer exponents.

What is the pattern in binomial expansions?

Key patterns in binomial expansion:

• There are n+1 terms
• The exponents of a decrease from n to 0
• The exponents of b increase from 0 to n
• The coefficients follow Pascal's triangle
• The sum of exponents in each term equals n

How do you find a specific term in a binomial expansion?

The kth term (where k starts at 0) is given by: T_k+1 = C(n,k) × a^n-k × b^k

For example, in (x + 2)⁵, the 3rd term (k=2) is: C(5,2) × x³ × 2² = 10 × 4x³ = 40x³