Proving Polynomial Identities
Aligned to A-APR.C.4 — Common Core State Standards for Mathematics.
What this lesson teaches
A polynomial identity is true for every value of the variable. You prove one by expanding both sides with algebra until they match, then you can use it to simplify numerical calculations.
Worked example
Prove (a + b)(a − b) = a^2 − b^2. Expand the left side: a×a − a×b + b×a − b×b = a^2 − ab + ab − b^2 = a^2 − b^2. Use it to compute 21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399.
Practice questions
- Prove (x + 1)^2 = x^2 + 2x + 1 by expanding.
- Use the identity a^2 − b^2 to compute 51 × 49.
- Verify (a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 for a = 1, b = 2.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Demostrar Identidades Polinómicas
Una identidad polinómica es verdadera para todo valor de la variable. Se demuestra desarrollando ambos lados con álgebra hasta que coincidan, y luego se puede usar para simplificar cálculos numéricos.
Ejemplo: Demuestra (a + b)(a − b) = a^2 − b^2. Desarrolla el lado izquierdo: a×a − a×b + b×a − b×b = a^2 − ab + ab − b^2 = a^2 − b^2. Úsala para calcular 21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399.
More Algebra lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required