Calculator Inputs
Enter coefficients from degree 5 down to degree 0 for both sides. The tool compares canonical polynomial forms and evaluates sample points.
Example Data Table
Example identity: (x^2 - 1)^2 = x^4 - 2x^2 + 1
| Expression Side | x^5 | x^4 | x^3 | x^2 | x | Constant |
|---|---|---|---|---|---|---|
| LHS = (x^2 - 1)^2 | 0 | 1 | 0 | -2 | 0 | 1 |
| RHS = x^4 - 2x^2 + 1 | 0 | 1 | 0 | -2 | 0 | 1 |
| Difference | 0 | 0 | 0 | 0 | 0 | 0 |
Formula Used
For a polynomial identity, define:
D(x) = L(x) - R(x)
If D(x) is the zero polynomial, then the identity is true for every value of x.
Coefficient method:
D(x) = c5x^5 + c4x^4 + c3x^3 + c2x^2 + c1x + c0
The identity holds exactly when:
c5 = c4 = c3 = c2 = c1 = c0 = 0
Numerical substitution helps illustrate equality at chosen points, but the coefficient comparison provides the full proof for polynomial forms entered in standard degree order.
How to Use This Calculator
- Write both polynomials in standard descending powers.
- Enter coefficients for the left side from x^5 to the constant term.
- Enter coefficients for the right side in the same order.
- Add sample x-values separated by commas for extra checking.
- Click Prove Identity.
- Read the difference polynomial and coefficient comparison.
- If every difference coefficient is zero, the identity is proved.
- Download the results as CSV or PDF if needed.
About This Calculator
This calculator helps students verify polynomial identities using a direct algebraic method. It turns both sides into comparable coefficient lists, builds the difference polynomial, checks several numerical points, and plots both expressions. It is useful for algebra classes, homework checking, and proof practice.
Polynomial identities are stronger than ordinary equations. An equation may hold for only some values, but an identity must hold for every value of the variable. That is why this tool emphasizes the zero-difference polynomial. When the entire difference collapses to zero, the proof is complete.
The graph adds another layer of understanding. If the left and right expressions are identical, both curves overlap exactly and the difference curve stays on the horizontal axis. This visual confirmation helps learners connect symbolic manipulation with function behavior in a clear way.
FAQs
1. What does this calculator prove?
It checks whether two entered polynomials are identical for every x-value. It proves this by comparing coefficients of the difference polynomial, not only by testing a few sample numbers.
2. Why is coefficient comparison enough?
If two polynomials are equal for all x, their difference must be the zero polynomial. A zero polynomial has every coefficient equal to zero, so coefficient matching gives a complete proof.
3. Do numerical test points prove the identity?
No. Test points are supportive checks only. A polynomial can match at some selected values and still fail elsewhere. The coefficient comparison is the decisive part of the proof.
4. What degree does this version support?
This page supports polynomials up to degree five on each side. You can still enter lower-degree expressions by placing zeros in higher-degree coefficient boxes.
5. What if the difference polynomial is not zero?
Then the expressions are not an identity. The calculator will show the remaining terms and the numerical table will usually display nonzero differences for some test values.
6. Can I use decimals and negative coefficients?
Yes. The calculator accepts positive, negative, and decimal coefficients. This is helpful for general algebra work, modeling exercises, and checking transformed expressions.
7. Why does the graph matter?
The graph offers visual insight. If both expressions are identical, their curves overlap completely and the difference graph stays on zero across the plotted interval.
8. When is this tool most useful?
It is useful while expanding brackets, factoring, rearranging identities, preparing notes, or checking homework. It helps confirm algebraic work step by step with tables and graphs.