Calculator Inputs
Example data table
| First base | Second base | a³ | b³ | a³ − b³ | (a − b)(a² + ab + b²) |
|---|---|---|---|---|---|
| 5 | 2 | 125 | 8 | 117 | 3 × 39 = 117 |
| 4 | 1 | 64 | 1 | 63 | 3 × 21 = 63 |
| 3.5 | 1.5 | 42.875 | 3.375 | 39.5 | 2 × 19.75 = 39.5 |
| 7 | 7 | 343 | 343 | 0 | 0 × 147 = 0 |
Formula used
Core identity: a3 − b3 = (a − b)(a2 + ab + b2)
First cube: a3 = a × a × a
Second cube: b3 = b × b × b
Verification: Multiply the two factors and compare the product with a3 − b3.
How to use this calculator
- Enter the first cube base for the first term.
- Enter the second cube base for the second term.
- Choose symbol labels if you want a custom symbolic display.
- Select the number of decimal places for formatted output.
- Set graph start, graph end, and graph points.
- Press the calculate button to show the result section.
- Review the symbolic form, numeric evaluation, factorization, and graph.
- Use the export buttons to save the current result as CSV or PDF.
FAQs
1) What does the calculator factor?
It factors and evaluates expressions of the form a³ − b³. The tool shows the identity, computes each cube, verifies the product of factors, and plots a related curve.
2) What is the difference of cubes identity?
The identity is a³ − b³ = (a − b)(a² + ab + b²). It rewrites a cube difference into one linear factor and one quadratic factor.
3) Can I use negative numbers?
Yes. Negative inputs are valid. The calculator cubes them correctly, applies the same identity, and still checks the factorized product against the direct subtraction.
4) Why does the graph use y = x³ − b³?
The graph keeps the second base fixed and varies x. This makes the algebra easier to visualize, shows where the curve crosses zero, and highlights the evaluated first-base point.
5) What does the root mean here?
The root is the x-value where y = x³ − b³ equals zero. That happens when x = b, because x³ and b³ become equal and their difference becomes zero.
6) Why is there a verification error field?
It measures the difference between direct subtraction and the factored-product result. With ordinary inputs, it should be zero or extremely close to zero because of rounding.
7) Are the symbol labels used in calculation?
No. Symbol labels only change the displayed algebraic expression. Numeric results always come from the entered first base, second base, and the chosen formatting settings.
8) When is the result zero?
The result is zero when the two cube bases are equal. In that case, a³ and b³ match exactly, and the linear factor a − b also becomes zero.