Proving Polynomial Identities
Aligned to A-APR.C.4 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. A polynomial identity is an equation true for every value of the variable. Prove one by expanding each side to matching form. Identities like (x + y)^2 = x^2 + 2xy + y^2 also describe numerical patterns.
Worked example
Prove (x + y)^2 = x^2 + 2xy + y^2. Expand the left side: (x + y)(x + y) = x^2 + xy + xy + y^2 = x^2 + 2xy + y^2. Both sides match, so it is an identity.
Practice questions
- Expand (x − y)^2 to show it equals x^2 − 2xy + y^2.
- Verify the identity (x + 3)^2 = x^2 + 6x + 9.
- Prove the difference-of-squares identity (x + y)(x − y) = x^2 − y^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
Este es un estándar avanzado (+). Una identidad polinómica es una ecuación verdadera para todo valor de la variable. Se demuestra desarrollando cada lado hasta una forma que coincida. Identidades como (x + y)^2 = x^2 + 2xy + y^2 también describen patrones numéricos.
Ejemplo: Demuestra (x + y)^2 = x^2 + 2xy + y^2. Desarrolla el lado izquierdo: (x + y)(x + y) = x^2 + xy + xy + y^2 = x^2 + 2xy + y^2. Ambos lados coinciden, así que es una identidad.
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