\displaystyle \textbf{H.C.F. of Algebraic Expressions}

\displaystyle \text{The H.C.F. of two or more algebraic expressions is the}
\displaystyle \text{greatest algebraic expression that divides each of them exactly,}
\displaystyle \text{that is, without leaving a remainder.}

\displaystyle \text{For example, consider }18x^2y\text{ and }12xy^2.
\displaystyle \text{H.C.F. of the numerical coefficients }18\text{ and }12=6.
\displaystyle \text{The common variables with their lowest powers are }x^1y^1.
\displaystyle \therefore \text{ H.C.F.}=6xy.

\displaystyle \frac{18x^2y}{6xy}=3x,\qquad \frac{12xy^2}{6xy}=2y.
\displaystyle \text{Thus, }6xy\text{ divides both expressions exactly.}
\displaystyle \\

\displaystyle \textbf{L.C.M. of Algebraic Expressions}

\displaystyle \text{The L.C.M. of two or more algebraic expressions is the}
\displaystyle \text{algebraic expression of the least degree that is exactly}
\displaystyle \text{divisible by each of the given expressions.}

\displaystyle \text{For example, consider }18x^2y\text{ and }12xy^2.
\displaystyle \text{L.C.M. of the numerical coefficients }18\text{ and }12=36.
\displaystyle \text{The highest powers of the variables are }x^2\text{ and }y^2.
\displaystyle \therefore \text{ L.C.M.}=36x^2y^2.

\displaystyle \frac{36x^2y^2}{18x^2y}=2y,\qquad \frac{36x^2y^2}{12xy^2}=3x.
\displaystyle \text{Thus, both given expressions divide the L.C.M. exactly.}
\displaystyle \\

\displaystyle \textbf{H.C.F. of Monomials}

\displaystyle \text{The H.C.F. of two or more monomials is the product of the}
\displaystyle \text{H.C.F. of their numerical coefficients and the common literals}
\displaystyle \text{taken with their lowest powers.}

\displaystyle \textbf{Method of Finding the H.C.F. of Monomials}

\displaystyle \text{H.C.F. of monomials}
\displaystyle =\text{H.C.F. of numerical coefficients}\times \text{H.C.F. of literals}.

\displaystyle \text{Example: Find the H.C.F. of }18x^2y^3 \text{ and }24x^3y^4z.

\displaystyle \text{H.C.F. of }18\text{ and }24=6.
\displaystyle \text{Lowest common powers are }x^2\text{ and }y^3.
\displaystyle \therefore \text{ H.C.F.}=6x^2y^3.

\displaystyle \text{Note: Since }z\text{ does not occur in the first monomial,}
\displaystyle \text{it is not a common factor and is therefore not included.}
\displaystyle \\

\displaystyle \textbf{L.C.M. of Monomials}

\displaystyle \text{The L.C.M. of two or more monomials is the product of the}
\displaystyle \text{L.C.M. of their numerical coefficients and all the literals}
\displaystyle \text{taken with their highest powers.}

\displaystyle \textbf{Method of Finding the L.C.M. of Monomials}

\displaystyle \text{L.C.M. of monomials}
\displaystyle =\text{L.C.M. of numerical coefficients}\times \text{L.C.M. of literals}.

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

\displaystyle \text{L.C.M. of }18\text{ and }24=72.
\displaystyle \text{Highest powers are }x^3,\ y^4\text{ and }z^1.
\displaystyle \therefore \text{ L.C.M.}=72x^3y^4z.
\displaystyle \\

\displaystyle \textbf{H.C.F. and L.C.M. of Polynomials}

\displaystyle \text{To find the H.C.F. and L.C.M. of polynomials, first factorize}
\displaystyle \text{each polynomial completely into its numerical and algebraic factors.}

\displaystyle \text{H.C.F.}= \text{H.C.F. of numerical coefficients}\times \text{common factors with lowest powers}.

\displaystyle \text{L.C.M.}= \text{L.C.M. of numerical coefficients}\times \text{all factors with highest powers}.

\displaystyle \text{Example: Find the H.C.F. and L.C.M. of } x^2+xy\text{ and }x^2-y^2.

\displaystyle x^2+xy=x(x+y).
\displaystyle x^2-y^2=(x+y)(x-y).

\displaystyle \text{The common factor is }(x+y).
\displaystyle \therefore \text{ H.C.F.}=x+y.

\displaystyle \text{For the L.C.M., take each factor with its highest power.}
\displaystyle \therefore \text{ L.C.M.}=x(x+y)(x-y).
\displaystyle \\

\displaystyle \textbf{Reducing an Algebraic Fraction to its Lowest Form}

\displaystyle \text{An algebraic fraction is in its simplest form, or lowest terms,}
\displaystyle \text{when its numerator and denominator have no common factor}
\displaystyle \text{other than }1.

\displaystyle \text{To reduce an algebraic fraction to its lowest form, factorize}
\displaystyle \text{the numerator and denominator completely and cancel the}
\displaystyle \text{common factors, while retaining the restrictions on the variable.}

\displaystyle \text{Example: Simplify } \frac{3x^2-12}{x^2-3x-10}.

\displaystyle \frac{3x^2-12}{x^2-3x-10} =\frac{3(x^2-4)}{(x-5)(x+2)}

\displaystyle =\frac{3(x+2)(x-2)}{(x-5)(x+2)}

\displaystyle =\frac{3(x-2)}{x-5},\qquad x\ne-2,\ 5.
\displaystyle \\

\displaystyle \textbf{Simplification of Expressions Involving Algebraic Fractions}

\displaystyle \text{An algebraic fraction is a fraction whose numerator or}
\displaystyle \text{denominator, or both, are algebraic expressions.}

\displaystyle \text{For example, }\frac{3x^2-12}{x^2-3x-10} \text{ is an algebraic fraction.}

\displaystyle \text{Expressions involving algebraic fractions can be simplified}
\displaystyle \text{using the same basic principles used for numerical fractions.}

\displaystyle \text{Factorize first, cancel only common factors, and always ensure}
\displaystyle \text{that the denominator is not equal to zero.}
\displaystyle \\

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