The Remainder Theorem
Aligned to A-APR.B.2 — Common Core State Standards for Mathematics.
What this lesson teaches
When a polynomial p(x) is divided by (x − a), the remainder equals p(a). So you can find the remainder just by evaluating the polynomial at a, without doing full division.
Worked example
Find the remainder when p(x) = x^3 − 2x^2 + 5x − 1 is divided by (x − 2). By the theorem, the remainder is p(2) = 8 − 8 + 10 − 1 = 9. So dividing leaves a remainder of 9.
Practice questions
- Find the remainder when p(x) = x^2 + 3x − 4 is divided by (x − 1).
- Use the theorem to find the remainder of x^3 − 8 divided by (x + 2).
- Is (x − 3) a factor of x^2 − 9? Explain using p(3).
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
El Teorema del Residuo
Cuando un polinomio p(x) se divide entre (x − a), el residuo es igual a p(a). Así puedes encontrar el residuo solo evaluando el polinomio en a, sin hacer la división completa.
Ejemplo: Encuentra el residuo cuando p(x) = x^3 − 2x^2 + 5x − 1 se divide entre (x − 2). Por el teorema, el residuo es p(2) = 8 − 8 + 10 − 1 = 9. Así que dividir deja un residuo de 9.
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