CBSE Class 8 Mathematics Algebraic Expressions and Factorisation Notes Set 04

Factors and Factorisation

When an expression is written as a product of two or more expressions, each of them is a factor. Writing an expression as such a product is called factorisation, and it undoes expansion.

Identities to Remember

\( (a+b)^2=a^2+2ab+b^2 \), \( (a-b)^2=a^2-2ab+b^2 \), \( a^2-b^2=(a+b)(a-b) \).
\( a^3+b^3=(a+b)(a^2-ab+b^2) \), \( a^3-b^3=(a-b)(a^2+ab+b^2) \).
\( a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca) \), so if a + b + c = 0 then \( a^3+b^3+c^3=3abc \).

Method 1: Common Monomial Factor

Find the HCF of all terms, divide each term by it, and write HCF times the quotients. Example: 3x²y + 9xy² - 15xy = 3xy(x + 3y - 5).

Method 2: Common Binomial or Grouping

If a bracket appears in every term, take it out: 2a(x + y) - 3b(x + y) = (x + y)(2a - 3b). If no factor is common, group terms in pairs: x³ + x² + x + 1 = x²(x + 1) + 1(x + 1) = (x + 1)(x² + 1).

Method 3: Identities

Difference of squares: 1 - (x - y)² = (1 + x - y)(1 - x + y). Perfect squares: x² + 49y² - 14xy = (x - 7y)². Perfect cubes and sums or differences of cubes use the cube identities, for example a³ - 18a² + 108a - 216 = (a - 6)³.

Method 4: Splitting the Middle Term

To factorise x² + px + q find two numbers with sum p and product q. For ax² + bx + c, find two numbers with sum b and product ac and split the middle term. Example: 15x² - 26x + 8: 15 × 8 = 120 and -20 - 6 = -26, so 15x² - 20x - 6x + 8 = 5x(3x - 4) - 2(3x - 4) = (3x - 4)(5x - 2).

Checking and Dividing

Always check by multiplying the factors. To divide an expression by a factor, factorise and cancel: (x² + 6x + 9) ÷ (x + 3) = x + 3. If the remainder on division is not zero the divisor is not a factor.

Worked Example

Factorise 6x² + 17x + 12.
6 × 12 = 72 and 8 + 9 = 17, so 6x² + 8x + 9x + 12 = 2x(3x + 4) + 3(3x + 4) = (2x + 3)(3x + 4).