ARCH-MM-13: Binomial Expansion (a + bx)^n Term Isolation, Constant Terms & Multi-Bracket Coefficients

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-13: Why the 150s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~150s (Blows 90s Clock)

Writing out the binomial expansion terms up to

r = 3

takes ~135s. The general term is

T_r = \binom{6}{r}(2)^{6-r}(-1)^r x^{1(6-r) - 1r}

. Setting the exponent to

0

gives

r = 3

, yielding

20 \times 8 \times (-1) = -160

The Fast ShortcutTarget: 40s (Saves 110s)

Line 1 (Zero-Exponent Index): Solve

1(6 - r) = 1r \implies r = 3

in 5s.
Line 2 (Isolate Single Term):

\binom{6}{3}(2)^{3}(-1)^{3} = 20 \times 8 \times (-1) = -160

(30s).

Where Students Lose Time: In 34 past UAT-UK questions (8.29% of Mathematics 2), candidates lose 110s by treating the minus sign as positive for odd powers r in ncr * a^(n-r) * (-bx)^r. Instead, apply MM2 with Maths 1 skill M4.1.

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