source: ncert
topic: algebra
class_level: class_8
chapter: ch9
difficulty: intermediate
Class 8 | Ch9: Algebraic Expressions and Identities MULTIPLICATION OF ALGEBRAIC EXPRESSIONS: Monomial × Monomial: multiply coefficients, add powers. 3x^2 × 4x^3=12x^5. Monomial × Polynomial: distribute. 2x(3x^2-5x+1)=6x^3-10x^2+2x. Polynomial × Polynomial: each term of first × each term of second. STANDARD ALGEBRAIC IDENTITIES: Identity 1: (a+b)^2 = a^2+2ab+b^2 Identity 2: (a-b)^2 = a^2-2ab+b^2 Identity 3: (a+b)(a-b) = a^2-b^2 Identity 4: (x+a)(x+b) = x^2+(a+b)x+ab USING IDENTITIES FOR CALCULATIONS: 105^2=(100+5)^2=10000+1000+25=11025. 98×102=(100-2)(100+2)=10000-4=9996. FACTORISATION USING IDENTITIES: x^2+8x+16=(x+4)^2. Check: 2×4=8, 4^2=16. Correct. 9a^2-12ab+4b^2=(3a-2b)^2. Check: 2×3a×2b=12ab. Correct. x^2-25=(x+5)(x-5). (a^2-b^2 type) SOLVED EXAMPLES: Example 1: Expand (3x+2y)^2=9x^2+12xy+4y^2. Example 2: Expand (4a-3b)^2=16a^2-24ab+9b^2. Example 3: (2x+3)(2x+5)=4x^2+(3+5)2x+15=4x^2+16x+15. Using identity 4: (x+a)(x+b) with 2x,3,5. Example 4: Evaluate 97^2=(100-3)^2=10000-600+9=9409. COMMON MISTAKES: (a+b)^2=a^2+2ab+b^2 NOT a^2+b^2 (missing middle term 2ab). (a-b)^2=a^2-2ab+b^2 NOT a^2-2ab-b^2 (last term is always positive). Factorisation: must check using expansion.