Algebra
Vieta's Formulas shortcut — For quadratic ax²+bx+c: sum of roots = -b/a, product = c/a. For cubic: sum=-b/a, sum of products of pairs=c/a, product=-d/a. Memorize as "alternating signs, descending powers."
Sophie Germain Identity — a⁴+4b⁴ = (a²+2b²+2ab)(a²+2b²-2ab). Useful for factoring fourth powers. Remember: "add 4 times fourth power, split middle 2ab term with ± signs."
Difference of powers factorization — aⁿ-bⁿ always divisible by (a-b). When n is odd, also divisible by (a+b). Quick check: odd powers split both ways, even powers only one way.
AM-GM application trigger — When you see products to maximize or sums to minimize with constant product/sum, apply AM≥GM. For positive reals: (a+b)/2 ≥ √(ab), equality when a=b.
Binomial and Multinomial Theorems
Binomial coefficients symmetry — ⁿCᵣ = ⁿCₙ₋ᵣ. Reading Pascal's triangle from either end gives same values. Use this to simplify: if r > n/2, compute ⁿCₙ₋ᵣ instead for faster calculation.
Middle term identification — For (a+b)ⁿ: if n even, middle term is (n/2 + 1)th term. If n odd, two middle terms at positions (n+1)/2 and (n+3)/2. Largest coefficient always at middle.
Binomial sum trick — Sum of all binomial coefficients: ⁿC₀+ⁿC₁+...+ⁿCₙ = 2ⁿ. Put x=1 in (1+x)ⁿ. For alternating sum (odd positions minus even): put x=-1, equals 0.
Complex Numbers
i powers cycle: i-LOOP — i¹=i, i²=-1, i³=-i, i⁴=1, then repeats. For iⁿ, divide n by 4: remainder 0→1, 1→i, 2→-1, 3→-i. Four-step cycle only.
De Moivre's formula — (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Remember: "power n outside becomes multiplier n inside the angle." Essential for roots of unity.
Conjugate root theorem — If polynomial has real coefficients and complex root a+bi, then a-bi is also root. They come in conjugate pairs; use to reduce degree quickly.
|z₁z₂| = |z₁||z₂| — Modulus of product equals product of moduli. Similarly |z₁/z₂| = |z₁|/|z₂|. But arg(z₁z₂) = arg(z₁) + arg(z₂). Modulus multiplies, argument adds.
Inequalities
Cauchy-Schwarz quick form — (a₁²+a₂²+...)(b₁²+b₂²+...) ≥ (a₁b₁+a₂b₂+...)². Equality when sequences are proportional. Remember: "sum of squares times sum of squares ≥ square of sum of products."
Triangle inequality — |a+b| ≤ |a|+|b| and |a-b| ≥ ||a|-|b||. For complex numbers and vectors. Visualize: direct path never exceeds sum of two sides; difference at least equals difference of magnitudes.
Rearrangement inequality — If a₁≤a₂≤...≤aₙ and b₁≤b₂≤...≤bₙ, maximum sum when similarly ordered, minimum when oppositely ordered. Match largest with largest for maximum.
Polynomials
Rational Root Theorem — For polynomial with integer coefficients, rational root p/q must have: p divides constant term, q divides leading coefficient. Test only these limited candidates systematically.
Remainder theorem shortcut — When P(x) divided by (x-a), remainder is P(a). For (ax-b), remainder is P(b/a). Faster than long division for finding remainders.
Descartes' Rule of Signs — Count sign changes in P(x) for positive roots; changes in P(-x) for negative roots. Number of roots equals sign changes or less by even number.
Number System
Divisibility by 9 and 11 — Divisible by 9: sum of digits divisible by 9. Divisible by 11: alternating sum (odd positions minus even positions) divisible by 11. Digit manipulation only.
Last digit patterns — Powers' last digits cycle: 2(4-cycle), 3(4-cycle), 4(2-cycle), 7(4-cycle), 8(4-cycle), 9(2-cycle). Divide exponent by cycle length, use remainder to find last digit.
Modular arithmetic trick — (a+b) mod m = ((a mod m)+(b mod m)) mod m. Similarly for multiplication. Reduce at each step to avoid large numbers in computations.