Polynomial Remainder Theorem
Polynomials · Class X
Divide p(x) = x^3 + bx^2 + cx + d by (x - k) and read off the remainder p(k).
- p(k) = remainder0
- Is (x - k) a factor?1
- p(0)-2
- y = p(x)
Formulas in this lab
- p(k) = remainder
- Is (x - k) a factor?
- p(0)
Frequently asked questions
▶What does the remainder theorem say?
When a polynomial $p(x)$ is divided by $(x - k)$, the remainder is simply $p(k)$. So no long division is needed for the remainder alone. If $p(k) = 0$, then $(x - k)$ is a factor of $p(x)$.
▶How do I use this lab?
Set b, c, d to fix the cubic $p(x) = x^3 + bx^2 + cx + d$, then pick k. The lab shows $p(k)$ and confirms it equals the remainder. Try $p(x) = x^3 - 4x^2 + x + 6$ with k = 2 to see remainder 0.
▶Common mistake on this topic
Students perform full long division when only the remainder is asked, wasting precious exam time. Use the remainder theorem: substitute and finish. Mind the sign: for $(x + 3)$ use $k = -3$.
▶Where is this used outside textbooks?
Engineers checking error patterns in digital data use polynomial remainders (CRC codes) to detect transmission errors. Inside the lab, the remainder gives instant feedback on whether a candidate is a root.