MathematicsChapter 4: Factorization and Algebraic Manipulation

Hcf, Lcm, and Algebraic Fractions

For algebraic expressions, factorise first.

Definition

For algebraic expressions, factorise first. HCF is the product of common factors with the lowest powers. LCM is the product of all factors with the highest powers. Then algebraic fractions cancel common factors (never cancel terms across a plus).

Overview

Two jobs that look similar:

Hcf and Lcm

HCF of 6 x^2 y and 9 x y^2 is 3 x y. LCM is 18 x^2 y^2. For quadratics, factor, then pick common brackets for HCF and all brackets for LCM.

Examples HCF of (x-1)(x+2) and (x-1)(x+5) is (x-1) · LCM adds the extra brackets · Numbers: 12 and 18 -> HCF 6, LCM 36

Fractions

x²-1x-1 = x+1 for x != 1, because x²-1=(x-1)(x+1). You may not cancel the 1s in x+1x. State excluded values where a denominator is zero.

Examples (x^2-9)/(x-3) = x+3 (x!=3) · Do not cancel x in (x+2)/x to get 2 · LCD to add fractions

Explanation

Square roots of algebraic squares: sqrt(x^2) is |x| in a careful course; many Class 9 papers still write x with a domain note. Follow the book, but never sqrt(a^2+b^2) = a+b.

In simple words

HCF x LCM = product of the two expressions (up to sign) when you work with monomials, same as for numbers. Use it as a check.

Key points

factorise first
lowest vs highest powers
cancel factors only
common lowest
all highest

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.