Calculator Inputs
Formula Used
General binomial theorem:
(ax ± by)n = Σ C(n, k) (ax)n−k (±by)k, for k = 0 to n.
nth term rule: Tk+1 = C(n, k) (ax)n−k (±by)k
Combination
C(n, k) = n! / [k!(n − k)!]. This gives the coefficient count pattern seen in Pascal’s triangle.
Sign behavior
A plus sign keeps all terms positive when coefficients are positive. A minus sign alternates signs across odd and even powers.
Evaluation
After expansion, each term is evaluated using the chosen variable values. The total is the sum of all evaluated terms.
How to Use This Calculator
1. Enter coefficients
Type the first and second numeric coefficients. The second coefficient is treated with the chosen plus or minus operator.
2. Name variables
Set variable symbols such as x and y. Use different names so the output stays readable and term powers remain separated.
3. Choose exponent and term
Select the exponent n and choose any term number to highlight. The table will emphasize that exact term.
4. Expand and export
Press the button to view the expansion, graph, and evaluation. Then download CSV or PDF whenever needed.
Example Data Table
| Expression | Exponent | Expanded Result | Value at 1,1 |
|---|---|---|---|
| (2x + 3y)3 | 3 | 8x3 + 36x2y + 54xy2 + 27y3 | 125 |
| (3a − b)4 | 4 | 81a4 − 108a3b + 54a2b2 − 12ab3 + b4 | 16 |
| (x + y)5 | 5 | x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 | 32 |
FAQs
1. What does this calculator expand?
It expands expressions of the form (a·variable₁ ± b·variable₂)^n. It shows the full polynomial, selected term, evaluation using chosen values, and a term-by-term coefficient table.
2. Why is the exponent limited?
Very large exponents create long outputs and heavy coefficient growth. Limiting the exponent keeps the page fast, the expansion readable, and exported files easier to review.
3. What is the highlighted term number?
The term number selects one expansion term for emphasis. For example, term 1 is the leading term, while term n+1 is the final term.
4. How are negative signs handled?
When you choose minus, the second term becomes negative inside the binomial. Odd powers of that term stay negative, while even powers become positive.
5. What does the graph show?
The Plotly chart compares numeric coefficients with evaluated term values. This helps you see magnitude changes, symmetry, sign alternation, and which terms dominate numerically.
6. What is the coefficient sum?
It is the sum of all numeric coefficients in the expansion. You can also get it by setting both variables to 1 before evaluating.
7. Why should variable names be different?
Different names keep the algebra visually clear. If both names match, powers merge into one variable and the displayed structure becomes less informative for learning.
8. What is included in CSV and PDF exports?
CSV includes the expression summary and the full term table. PDF captures the visible result section, including the expansion summary and the rendered chart.