ALGEBRA DEEP DIVE
⏱ 12 Min Read • Step-by-Step Worked Proofs
Mastering the Remainder Theorem: Synthetic Division vs. Polynomial Long Division
Why the Remainder Theorem is the fastest shortcut on the SAT and calculus exams to factor high-degree polynomials without tedious long division. We break down the algebraic proof, synthetic table setups, and common negative sign traps.
functions
The Fundamental Identity
LaTeX Verified
P(x) = (x - c) · Q(x) + R ⟹ P(c) = R
// If P(c) = 0, then (x - c) is an exact factor of P(x).
SC
Dr. Sarah Chen
Computational Mathematics
Worked Interactive Step-Proof
c = 3
// Divide 2x³ - 5x² + 3x - 7 by (x - 3)
3 |
2
-5
3
-7
↓
6
3
18
2
1
6
R = 11
Check via direct substitution:
P(3) = 2(27) - 5(9) + 3(3) - 7 = 54 - 45 + 9 - 7 = 11. Remainder matches in 10 seconds without polynomial long division!
Topic: Polynomial Algebra
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