ARCH-MM-01: Polynomial Division by a Quadratic & Linear Remainder Theorem (R(x) = mx + n)

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-01: Why the 165s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~165s (Blows 90s Clock)

Performing polynomial long division of

x^{3} + 2 x^{2} - 5 x + 4

x^{2} - 5 x + 6

step by step yields quotient

Q(x)

and linear remainder

R(x) = 24x - 38

, taking ~150s with high risk of sign errors.

The Fast ShortcutTarget: 45s (Saves 120s)

Line 1 (Linear Remainder Identity): Factor

x^{2} - 5 x + 6 = (x - 2)(x - 3)

; write

P(2) = 2m + n = 10

and

P(3) = 3m + n = 34

.
Line 2 (Instant Subtraction):

m = \frac{34 - (10)}{3 - (2)} = 24

and

n = -38

, giving

-14

(30s).

Where Students Lose Time: In 20 past UAT-UK questions (4.88% of Mathematics 2), candidates lose 120s by forgetting that dividing by degree-2 divisor (x-a)(x-b) leaves a degree-1 remainder mx+n. Instead, apply MM1 with Maths 1 skill M4.2.

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