H.C.F. of Algebraic Expression: The H.C.F. of two or more algebraic expressions is an expression of the highest degree which divides each of them without leaving any remainder.

For example, the H.C.F of $18x^2 y \text{ and } 12xy^2$ is $6xy.$

If you divide $18x^2 y \text{ by } 6xy,$ we see there is no remainder. Quotient $=3x$

L.C.M. of Algebraic Expression: The L.C.M of two or more algebraic expressions is an expression of the lowest degree which is divisible by each of the given expressions without leaving any remainder.

For example, the L.C.M of $18x^2 y \text{ and } 12xy^2$ is $72x^2 y^2$

So if you divide $72x^2 y^2 \text{ by } 18x^2 y$ we get $4y$ . No remainder.

H.C.F. of Monomials: The H.C.F. of two or more monomials is the product of the H.C.F. of their numerical coefficient, and the H.C.F. of the Literals (which is the lowest power of each of the literals).

Method of Finding the H.C.F. of Monomials

H.C.F. of given monomials = H.C.F. of numerical coefficients × H.C.F. of literals

Let’s do an example:Find H.C.F of $18x^2 y^3 \text{ and } 24x^3 y^4 z$

H.C.F=H.C.F of numerical coefficients ×H.C.F of literals = $6x^2 y^3$

Note: there is no $z$ in the first monomial and hence there is no z in the

L.C.M of Monomials: The L.C.M. of two or more monomials is the product of the L.C.M. of their numerical coefficient, and the L.C.M of the Literals (which is the highest power of each of the literals).

Method of finding L.C.M of Monomials

L.C.M of given monomials = L.C.M. of numerical coefficients × L.C.M of literals

Example:Find L.C.M of $18x^2 y^3 \text{ and } 24x^3 y^4 z$

L.C.M=L.C.M of numerical coefficients ×L.C.M of literals $=72\times x^3 y^4 z$

Note:The highest power of $x, y \text{ and } z \text{ are } x^3,y^4 and z^1$

HCF and LCM of Polynomials: For finding the H.C.F. and L.C.M. of given polynomials, we factorize each of them and evaluate: $\text{ C.F. = (H.C.F. of Numerical Coefficients)} \times \text{(Each common factor raised to lowest power) }$ $\text{ C.M = (L.C.M. of Numerical Coefficients) } \times \text{(Each factor raised to highest power) }$

Example:

Lets find the H.C.F and L.C.M of $x^2+xy \text{ and } x^2-y^2$

First factorize the ploynomials, we get $x^2+xy=x(x+y)$ $x^2-y^2=(x+y)(x-y)$

Therefore $H.C.F=(x+y)$ is common and the higest power is 1.

Hence the $H.C.F=(x+y)$

L.C.M=product of each factors, $x, (x+y), (x-y)=x(x+y)(x-y)$

Reducing and algebraic fraction into its lowest form: An algebraic fraction is said to be in its simplest form (or in lowest terms) if the numerator and denominator have no common factor (except 1), i.e., if the H.C.F. of the numerator and denominator is 1. $\displaystyle \text{Simplify: } \frac{3x^2-12}{x^2-3x-10}= \frac{3(x+2)(x-2)}{(x-5)(x+2)}=\frac{3(x+2)}{(x-5)}$

To reduce the algebraic fraction, it is easy. Factorize the numerator and the denominator and then cancel whatever is common. Let’s do one example to demonstrate that.

Simplification of Expressions Involving Algebraic Fractions

When you have a polynomial as numerator and a polynomial as its denominator is called an algebraic fraction. $\displaystyle \text{Example: } \frac{3x^2-12}{x^2-3x-10}$

Expressions involving algebraic fractions may be simplified in the same way as we simplify arithmetical expressions as demonstrated above.