This calculator factors expressions built from a shared base and a shared variable multiplier: E = Σ ci·bax+ri.
It extracts the greatest common coefficient, the smallest shared exponential offset, and a substitution view when offset gaps follow a repeatable step.
Calculator form
Example data table
| Example expression | Base | Shared multiplier | Common factor | Factored remainder |
|---|---|---|---|---|
| 12·32x+6 − 18·32x+2 + 6·32x+4 | 3 | 2 | 6·32x+2 | 2·34 + 32 − 3 |
| 8·2x+5 + 4·2x+3 | 2 | 1 | 4·2x+3 | 2·22 + 1 |
| 15·53n−1 − 10·53n−4 + 5·53n+2 | 5 | 3 | 5·53n−4 | 56 + 3·53 − 2 |
Formula used
Expression model: E = Σ ci·bax+ri
Greatest common coefficient: G = gcd(|c1|, |c2|, ...)
Smallest offset: rmin = min(ri)
Factored form: E = G·bax+rmin Σ (ci/G)·bri−rmin
Optional substitution: Let d = gcd(ri − rmin).
Then set t = bd and rewrite the remainder as a polynomial-like expression in t. This makes exponent spacing easier to inspect and compare.
How to use this calculator
- Choose how many terms appear in the exponential expression.
- Enter one shared base for all exponential terms.
- Enter the shared variable multiplier that appears in every exponent.
- Fill in each term’s integer coefficient and exponent offset.
- Set a variable value for the numeric check and graph range.
- Press Factor Expression to see the factored form, table, and Plotly graph above the form.
FAQs
1) What form of expression does this calculator support?
It supports sums or differences of exponential terms that share one base and one variable multiplier, such as 12·3^(2x+6) − 18·3^(2x+2) + 6·3^(2x+4).
2) Why must the terms share one base?
A shared base makes the common exponential factor valid. Without one base, the calculator cannot extract a single exponential part from every active term consistently.
3) What does the shared variable multiplier mean?
It is the coefficient on the variable inside every exponent. When that multiplier stays the same, the calculator can pull out the smallest offset cleanly.
4) What if one coefficient is zero?
Zero terms are ignored during factoring. This lets you keep the same form layout while effectively working with fewer active terms.
5) What is the substitution view used for?
It rewrites repeated offset spacing as a simpler variable, such as t = 3^2. That turns the remaining factor into a polynomial-like expression you can inspect faster.
6) Does the graph show the factored expression or the original one?
The graph plots each original term and the total expression across the selected variable range. This helps verify growth, decay, and agreement with the factored result.
7) Why is a numeric check included?
The numeric check evaluates both the original expression and its factored reconstruction at one variable value. Matching results confirm the factoring step is correct.
8) Can I export the results?
Yes. After factoring, you can download a CSV summary and a PDF snapshot that includes the result block, breakdown table, and graph.