The Remainder Theorem
Aligned to A-APR.B.2 — Common Core State Standards for Mathematics.
What this lesson teaches
For a polynomial p(x), the remainder when dividing by (x − a) equals p(a). So evaluating the polynomial at a quickly gives that remainder, and if p(a) = 0 then (x − a) is a factor.
Worked example
Let p(x) = x^2 − 5x + 6 and divide by (x − 2). Compute p(2) = 4 − 10 + 6 = 0, so the remainder is 0 and (x − 2) is a factor of p(x).
Practice questions
- Find the remainder when p(x) = x^2 + 3x − 4 is divided by (x − 1).
- Is (x + 2) a factor of p(x) = x^2 + x − 2? Use p(−2).
- Find p(3) for p(x) = x^3 − 2x^2 + x − 5 to get the remainder on division by (x − 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
Para un polinomio p(x), el residuo al dividir entre (x − a) es igual a p(a). Así, evaluar el polinomio en a da rápidamente ese residuo, y si p(a) = 0 entonces (x − a) es un factor.
Ejemplo: Sea p(x) = x^2 − 5x + 6 y divide entre (x − 2). Calcula p(2) = 4 − 10 + 6 = 0, así que el residuo es 0 y (x − 2) es un factor de p(x).
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