Polynomial Expansion
Expand (ax+b)^n using the binomial theorem, showing each term and coefficient.
About Polynomial Expansion
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)!)
Worked example
With the default values below, Polynomial Expansion returns Expanded Form of 1x^4 + 8x^3 + 24x^2 + 32x + 16. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (coefficient of x)
- 1
- b (constant term)
- 2
- n (exponent)accepts 0–20
- 4
Results
- Expanded Form
- 1x^4 + 8x^3 + 24x^2 + 32x + 16
- Number of Terms
- 5
Step by step
- 1Values used
- 2Binomial Theorem
- 3Expanded Form= 1x^4 + 8x^3 + 24x^2 + 32x + 16
- 4Number of Terms= 5
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.