The simplest tool to instantly expand any binomial expression $\mathbf{(x+y)^n}$ using the Binomial Theorem. Input the exponent (Power) and get the complete polynomial expansion.
Expansion of Binomial Calculator
Expansion of Binomial Calculator Formula
The Binomial Theorem is defined by the following equation:
$$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$
Where $\binom{n}{k}$ is the binomial coefficient, calculated as:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
Formula Source: Wikipedia: Binomial Theorem
Variables
The calculator uses a single variable, the exponent, to determine the length and coefficients of the resulting polynomial:
- Exponent ($\mathbf{n}$): The non-negative integer power to which the binomial expression $(\mathbf{x+y})$ is raised. The value of $\mathbf{n}$ determines the number of terms in the expansion ($\mathbf{n+1}$ terms).
- Term $\mathbf{x}$ and $\mathbf{y}$: These are placeholders for the two terms in the binomial. The calculator provides the expansion in terms of these two generic variables.
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What is Expansion of Binomial Calculator?
A binomial expansion calculator is a tool that applies the **Binomial Theorem** to find the polynomial result of raising a binomial (a two-term expression like $x+y$) to any non-negative integer power, $n$. Instead of manually multiplying the expression $n$ times, which can be tedious, the calculator instantly uses a formula based on combinations and exponents.
The core of the calculation relies on finding the **binomial coefficients** for each term. These coefficients can be read directly from Pascal’s Triangle or calculated using the combinations formula $\binom{n}{k}$, where $n$ is the power and $k$ is the term index (starting from 0). The theorem ensures that the sum of the exponents in every term always equals $n$.
How to Calculate Expansion of Binomial ($\mathbf{(x+y)^3}$ Example)
Here is the step-by-step process used by the calculator to expand $\mathbf{(x+y)^3}$:
- Identify $\mathbf{n}$ and $\mathbf{k}$ values: The exponent $\mathbf{n}$ is 3. The expansion will have $n+1 = 4$ terms, so $\mathbf{k}$ runs from 0 to 3.
- Calculate Binomial Coefficients $\mathbf{\binom{n}{k}}$:
- $k=0$: $\binom{3}{0} = 1$
- $k=1$: $\binom{3}{1} = 3$
- $k=2$: $\binom{3}{2} = 3$
- $k=3$: $\binom{3}{3} = 1$
- Determine the Exponents for $\mathbf{x}$ and $\mathbf{y}$: The power of $x$ starts at $n$ and decreases by one in each term, while the power of $y$ starts at 0 and increases by one.
- Term 1 ($k=0$): $x^{3-0}y^0 = x^3$
- Term 2 ($k=1$): $x^{3-1}y^1 = x^2y$
- Term 3 ($k=2$): $x^{3-2}y^2 = xy^2$
- Term 4 ($k=3$): $x^{3-3}y^3 = y^3$
- Combine Terms: Multiply the coefficient, $x$ term, and $y$ term for each $\mathbf{k}$ value and add them together.
- Term 1: $1 \cdot x^3 = x^3$
- Term 2: $3 \cdot x^2y = 3x^2y$
- Term 3: $3 \cdot xy^2 = 3xy^2$
- Term 4: $1 \cdot y^3 = y^3$
- Final Result: The expanded polynomial is $\mathbf{x^3 + 3x^2y + 3xy^2 + y^3}$.
Frequently Asked Questions (FAQ)
What is the Binomial Theorem used for?
The Binomial Theorem is fundamental in algebra, probability, and statistics. It provides a quick way to expand polynomials and is used extensively in deriving probability distributions (like the binomial distribution) and complex combinatorial analysis.
What is a binomial coefficient?
A binomial coefficient, denoted as $\binom{n}{k}$, represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to the order of selection. In the context of the theorem, it is the coefficient of each term in the expanded polynomial.
How does Pascal’s Triangle relate to the expansion?
Each row of Pascal’s Triangle corresponds exactly to the binomial coefficients $\binom{n}{k}$ for a given exponent $n$. For example, the row for $n=4$ is 1, 4, 6, 4, 1, which are the coefficients for the expansion of $(x+y)^4$.
Can I expand a binomial with negative exponents?
The standard Binomial Theorem applies only when the exponent $n$ is a non-negative integer. For negative or fractional exponents, a different concept called the **Generalized Binomial Theorem** must be used, which results in an infinite series.