# Binomial Theorem Expansion Calculator



An online expansion calculator of algebraic expressions is presented.

## Binomial Theorem

The binomial theorem [1 , 2] states that a polynomial of the form $(x + y)^n$ may be expaneded as a sum of terms in volving powers of $x$ and $y$ as follows: $(x + y)^n = \sum_{k=0}^{n} {n\choose k} (x)^{n-k} (y)^k$ where ${n\choose k}$ is called the binomial coefficient and is given by ${n\choose k} = \dfrac{n!}{k!(n-k)!}$
Example
Expand the expression $(x - 2y)^4$
Solution
Use the theorem above to write
$(x - 2y)^4 = (x + (-2y))^4 = \sum_{k=0}^{4} {4\choose k} (x)^{4-k} (-2y)^k \\ \quad = {4\choose 0} x^{4-0} (-2y)^0 + {4\choose 1}x^{4-1} (-2y)^1 + {4\choose 2}x^{4-2} (-2y)^2 + {4\choose 3}x^{4-3} (-2y)^3 + {4\choose 4}x^{4-4} (-2y)^4$

Calculate binomial coefficients
${4\choose 0} = \dfrac{4!}{0!(4)!} = 1 \qquad {4\choose 1} = \dfrac{4!}{1!3!} = 4 \\ {4\choose 2} = \dfrac{4!}{2!2!} = 6 \qquad {4\choose 3} = \dfrac{4!}{3!1!} = 4 \qquad {4\choose 4} = 1$

Substitute and simplify
$(x - 2y)^3 = x^4 + 4 x^3 (-2y) + 6 x^2 (-2y)^2 + 4 x (-2y)^3 + (-2y)^4 \\ \quad = x^4 - 8 x^3 y + 24 x^2 y^2 - 32 x y^3 + 16 y^4$

## Use of the Expansion Calculator

1 - Enter and edit the expression to expand and click "Enter Expression" then check what you have entered.
2 - The four operators used are: + (plus) , - (minus) , ^ (power) and * (multiplication). (example: (x - 2y)^4 )
2 - Click "Expand" to obain the expanded and simplified expression.

Expression:

Notes: In editing functions, use the following:
Here are some examples of expressions that you may copy and paste to practice:
(x+y)^2             (x - 4y)^2             (-3x-y)^2             (a-b)^3
(3a-2b)^3             (4x-5y)^4             (x+y)^5             (-2x+3y)^6

## More References and Links

Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8