Points to Remember: We Distribute, Yet Things Multiply | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

Distributive property

a(b + c) = ab + ac
(a + b)c = ac + bc
Example: 3(4 + 5) = 3×4 + 3×5 = 12 + 15 = 27

General product expansion (two-term each side)

(a + m)(b + n) = ab + an + bm + mn
Increase from ab = an + bm + mn
Example: (23 + 2)(27 + 3) = 23×27 + 23×3 + 2×27 + 2×3

One up, one down special cases

(a + 1)(b + 1) = ab + a + b + 1
(a + 1)(b − 1) = ab + b − a − 1
Example: (5 + 1)(7 + 1) = 35 + 5 + 7 + 1 = 48

Square of a sum / difference

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
Example: (6 + 5)² = 36 + 60 + 25 = 121

Difference of squares

(a + b)(a − b) = a² − b²
Example: 102 × 98 = (100 + 2)(100 − 2) = 100² − 2² = 10000 − 4 = 9996\\

Sum of two squares formula (useful pattern)

2(a² + b²) = (a + b)² + (a − b)²
Example: 2(6² + 5²) = (11)² + (1)² = 121 + 1 = 122

Cube-related pattern (binomial cube)

(a + b)(a² + 2ab + b²) = a³ + 3a²b + 3ab² + b³
(this is (a + b)³ expanded)
Example: (x + y)³ = x³ + 3x²y + 3xy² + y³

Useful algebra rules / tips

• Cross multiplication for proportion (appears elsewhere but handy):
  If a : b :: c : d then ad = bc.

• Like terms: only terms with identical variable parts (same letters and powers) can be added.
  Example: 2ab + 3ab = 5ab, but 3a² + 2a ≠ simplify further.

• Identity approach for fast mental arithmetic:
  a² = (a + b)(a − b) + b² — choose b so (a ± b) are easy to multiply.

Fast multiplication shortcuts (distributive ideas)

1. Multiply by 11 (digit-sum trick)
   To multiply a number by 11, write the number, then between every adjacent pair of digits put their sum (handle carries).
   Example: 3874 × 11 → digits: 3 8 7 4 → write: 3 (3+8) (8+7) (7+4) 4 = 3 11 15 11 4 → carry-adjust → 4 2 6 1 4 → 42614

2. Multiply by 101, 1001, 10001, ...
   Multiply by 100 + 1: insert one zero copy then add (for 101).
   Example: 3874 × 101 = 3874×100 + 3874 = 387400 + 3874 = 391274
   For 1001, insert two zeros (copy) and add, etc.

3. Multiply by numbers like 99, 999
   Use (100 − 1) etc.: 23478 × 999 = 23478 × (1000 − 1) = 23478000 − 23478

Area/geometry identities (common uses)

• Area of square of side (a + b) = (a + b)² = a² + 2ab + b²

• (m + n)² − 4mn = (n − m)² (useful for square-with-rectangles problems)

• Express complex areas as sums/differences using distributivity and simplify.

Handy problem-solving templates

1. If one factor increases by u and other by v:
   New product = ab + ub + av + uv

2. To split a total X in ratio m:n:
   First share = (m ÷ (m + n)) × X ; Second share = (n ÷ (m + n)) × X

3. Difference of two squares to compute near-squares quickly:
   a² = (a + b)(a − b) + b² — choose b small to ease multiplication.

4. Pattern translation (counting / tiles / steps):
   If you derive formulas like k(k + 2), k² + 2k, or (k + 1)² − 1, simplify to a single polynomial k² + 2k and use it for any k.