CBSE Class 8 Mathematics Algebraic Expressions and Factorisation Notes Set 03

What Is Factorisation?

To factorise an expression is to write it as a product of simpler expressions (its factors). It is the reverse of multiplying. Always check by multiplying the factors back.

Methods

Common factor: take out the greatest common factor of all the terms, e.g. 12x^3y^4 + 16x^2y^5 - 4x^5y^2 = 4x^2y^2(3xy^2 + 4y^3 - x^3).
Grouping: arrange the terms in pairs that share a common factor, e.g. x^2 + xy + 8x + 8y = x(x + y) + 8(x + y) = (x + y)(x + 8).
Difference of squares: a^2 - b^2 = (a + b)(a - b), e.g. 16x^4 - 81 = (4x^2 + 9)(2x + 3)(2x - 3).
Perfect squares: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2.
Splitting the middle term: x^2 + px + q is written with two numbers whose sum is p and product is q, e.g. x^2 - 8x - 20 = (x - 10)(x + 2).
Sum and difference of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Factor Theorem and Remainder

When p(x) is divided by (x - a) the remainder is p(a). If p(a) = 0 then (x - a) is a factor of p(x) (factor theorem).
Worked example: x^3 - 5x^2 + 8x - 4 vanishes at x = 1, so (x - 1) is a factor; dividing gives (x - 1)(x - 2)^2.

Division of Algebraic Expressions

Dividend = Divisor x Quotient + Remainder. For a monomial divisor divide each term by it. For a polynomial divisor use long division after arranging both in descending powers.