Algebraic Identities and Identity Checker
Review square and cube identities, expand both sides, and check whether polynomial expressions form an identity.
Check a polynomial identity
Try an example
Working steps
Common algebraic identities
| Identity | Equivalent expressions |
|---|---|
| Square of a sum | (a+b)^2 = a^2+2ab+b^2 |
| Square of a difference | (a-b)^2 = a^2-2ab+b^2 |
| Difference of squares | (a-b)(a+b) = a^2-b^2 |
| Cube of a sum | (a+b)^3 = a^3+3a^2b+3ab^2+b^3 |
| Cube of a difference | (a-b)^3 = a^3-3a^2b+3ab^2-b^3 |
| Sum of cubes | (a+b)(a^2-ab+b^2) = a^3+b^3 |
| Difference of cubes | (a-b)(a^2+ab+b^2) = a^3-b^3 |
Why expansion verifies a polynomial identity
An identity is true for every allowed value of its variables. Expanding and collecting both sides produces the same coefficient for each variable term. Their difference is therefore the zero polynomial. This is stronger than checking a few sample values.
For (x + 4)^2, expansion gives x² + 4x + 4x + 16, then x² + 8x + 16. The equation (x + 4)^2 = x^2 + 16 misses the 8x term, so it is not an identity. At x = 1 its two sides are 25 and 17. It happens to be true at x = 0, but one matching value does not prove an identity.
The checker compares polynomial coefficients exactly. When expressions differ, it displays their difference and a counterexample. “Not an identity” does not mean there are no values that make them equal; solving that equation is a separate task.
How identities help with factoring and mental math
Recognizing a special product can turn an expanded expression into useful factors. For x² − 49, the difference-of-squares pattern gives (x − 7)(x + 7). For x² + 10x + 25, the square-of-a-sum pattern gives (x + 5)².
The same patterns help with arithmetic. Compute 103² as (100 + 3)² = 10,000 + 600 + 9 = 10,609. Compute 48 × 52 as (50 − 2)(50 + 2) = 2,500 − 4 = 2,496. Keep the middle terms when squaring a sum; only the product of conjugate brackets cancels them.
The checker covers polynomial identities within its degree and size limits. Identities involving fractions need domain restrictions, and trigonometric identities require different methods. Those are outside this polynomial checker.
Further reading: OpenStax: polynomials and rational expressions.