Expand Each Binomial Calculator

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Expertly Reviewed by David Chen, CFA — Updated October 24, 2023

Mastering algebra requires precision. Use our expand each binomial calculator to instantly expand algebraic expressions of the form $(ax + b)^n$ using the Binomial Theorem. Whether for homework or advanced engineering, get accurate results and step-by-step solutions in seconds.

Expand Each Binomial Calculator

Enter values and click Calculate to see the expansion.

Expand Each Binomial Calculator Formula

$$(ax + b)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^{n-k} b^k$$

Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Formula Source: Wolfram MathWorld, Khan Academy

Variables:

  • a: The coefficient multiplied by the variable x.
  • b: The constant term inside the binomial.
  • n: The power or exponent to which the binomial is raised (non-negative integer).

What is Expand Each Binomial Calculator?

The expand each binomial calculator is a specialized algebraic tool designed to solve polynomial expansion problems. Instead of manually multiplying a binomial multiple times (e.g., $(x+1) \cdot (x+1) \cdot (x+1)$), it utilizes the Binomial Theorem to find the coefficient of every term efficiently.

This method is essential in statistics, calculus, and financial modeling where probability distributions like the Binomial Distribution require expanding complex power series. Using a calculator ensures no arithmetic errors occur during the factorial calculations of coefficients.

How to Calculate (Example)

Expanding $(2x + 3)^3$ step-by-step:

  1. Identify variables: $a=2, b=3, n=3$.
  2. Term 1 ($k=0$): $\binom{3}{0}(2x)^3(3)^0 = 1 \cdot 8x^3 \cdot 1 = 8x^3$.
  3. Term 2 ($k=1$): $\binom{3}{1}(2x)^2(3)^1 = 3 \cdot 4x^2 \cdot 3 = 36x^2$.
  4. Term 3 ($k=2$): $\binom{3}{2}(2x)^1(3)^2 = 3 \cdot 2x \cdot 9 = 54x$.
  5. Term 4 ($k=3$): $\binom{3}{3}(2x)^0(3)^3 = 1 \cdot 1 \cdot 27 = 27$.
  6. Combine: $8x^3 + 36x^2 + 54x + 27$.

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Frequently Asked Questions (FAQ)

What is the Binomial Theorem? It is a mathematical formula that provides the expansion of powers of a binomial.

Can I expand with negative constants? Yes, simply enter a negative value for “b” in the input field.

Is there a limit to the exponent? Our calculator supports exponents up to 20 to ensure browser stability and readability.

Why are coefficients called binomial coefficients? Because they represent the number of ways to choose terms during expansion, related to Pascal’s Triangle.

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