| --- |
| source: ncert |
| topic: algebra |
| class_level: class_8 |
| chapter: ch9 |
| difficulty: intermediate |
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| |
| 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. |
