Polynomial Expansion
Expand (ax+b)^n using the binomial theorem, showing each term and coefficient.
Inputs
Non-negative integer, max 20
Expanded Form
1x^4 + 8x^3 + 24x^2 + 32x + 16
Number of Terms
5
Step by step
Values used
a (coefficient of x) = 1; b (constant term) = 2; n (exponent) = 4
Binomial Theorem
(ax+b)^n = Σ C(n,k) · a^(n-k) · b^k · x^(n-k)
Expanded Form
= 1x^4 + 8x^3 + 24x^2 + 32x + 16
Number of Terms
= 5
How it works
The binomial theorem states that (ax+b)^n = Σ C(n,k)·(ax)^(n−k)·b^k for k = 0 to n. Each term's coefficient is a binomial coefficient C(n,k) multiplied by the appropriate powers of a and b.
Formula
Binomial Theorem
(ax+b)^n = Σ C(n,k) · a^(n-k) · b^k · x^(n-k)
- n
- Exponent
- k
- Term index
- C(n,k)
- Binomial coefficient n!/(k!(n-k)!)
Frequently Asked Questions
What is the maximum exponent supported?
The calculator supports exponents from 0 to 20. Beyond that, coefficients become extremely large.
Can I expand (x+1)^5?
Yes — set a=1, b=1, n=5 to get x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.