Subject: Compulsory Maths

HCF is the highest number that is the greatest thing for simplifying fractions. For example, the factors common to 12 and 3 are 2,3.

\(\therefore\) HCF of 12 and 8 is 3.

**HCF of monomial expressions**

We can find the HCF of the given monomial expressions just by taking the common variable with the least power. The HCF of the numerical coefficient is obtained as like in the case of arithmetic. For example,

Find the HCF of x^{4}y^{2}and x^{2}y^{4}

1st expression = x^{4}y^{2}

2nd expression = x^{2}y^{4}

\(\therefore\) HCF = x^{2}y^{2}**HCF of polynomial Expression**

We can find the HCF of polynomials by factorizing them. For example,

Find HCF of ax - bx and a2 - b2

1st expression = ax - bx= x(a-b)

2nd expression = a2 - b2 = (a-b)(a+b)

\(\therefore\) = (a-b)

**Lowest Common Multiple (LCM)**

The smallest positive number that the multiple of two or more numbers is LCM. For example, the LCM of a^{2} and a^{3} is a^{3}.

**Lowest common factor of monomial expressions**

We can find the LCM of the given monomial expressions just by taking the common variable with the highest power. For examples,

1sr expression = ax^{2}

2nd expression = a^{2}x^{2}

\(\therefore\) a^{2}x^{2}**LCM of polynomial expressions**

To find LCM of polynomial expressions . We should factorize them. Then the product of a common factors and remaining factors is the LCM of the given expressions. For examples,

Find the L.C.M of ax^{2}+ ax and a^{2}x^{2}+ a^{2}x

1st expression = ax^{2}+ ax = ax(x+1)

2nd expression = a^{2}x^{2}+ a^{2}x = a^{2}x(x+1)

\(\therefore\) LCM = a^{2}x(x+1)

- We can find the HCF of the given monomial expressions just by taking the common variable with the least power.
- We can find the HCF of polynomials by factorizing them.
- We can find the LCM of the given monomial expressions just by taking the common variable with the highest power.
- To find LCM of polynomial expressions . We should factorize them. Then the product of a common factors and remaining factors is the LCM of the given expressions.

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Find the HCF of x^{4}y^{2} and x^{2}y^{4}.

Solution:

1st expression = x^{4}y^{2}

2nd expression = x^{2}y^{4}

\(\therefore\) HCF = x^{2}y^{2}

Find the HCF of 4ab^{2} and 6a^{2}b^{3}.

Solution:

1st expression = 4ab^{2}

= 2×2ab^{2}

2nd expression = 6a^{2}b^{3}

= 2×3a^{2}b^{3}

\(\therefore\) HCF = 2ab^{2}

Find the HCF of ax - bx.

Solution:

1st expression = ax - bx = x(a-b)

2nd expression = a^{2} -b^{2} = (a+b)(a-b)

\(\therefore\) HCF = (a-b)

Find the HCF of x^{2} + xy + zx + yz and x^{2} - y^{2}.

Solution:

1st expression = x^{2} + xy + zx + yz = x(x+y) + z(x+y) = (x+y) (x+z)

2nd expression = x^{2} - y^{2} = (x+y) (x-y)

\(\therefore\) HCF = (x+y)

Find the LCM of ax^{2} and a^{2}x^{2}.

Solution:

1st expression = ax^{2}

2nd expression = a^{2}x^{2}

\(\therefore\) LCM = a^{2}x^{2}

Find the LCM of 4x^{3}y^{3} and 6xy.

Solution:

1st expression = 4x^{3}y^{3}

= 2×2x^{3}y^{3}

2nd expression = 6xy

= 2×3xy

\(\therefore\) LCM = 2×2×3 x^{3}y^{3}

= 12x^{3}y^{3}

Find the LCM of ax^{2} + ax and a^{2}x^{2} + a^{2}x.

Solution:

1st expression = ax^{2} + ax = ax(x+1)

2nd expression = a^{2}x^{2} + a^{2}x = a^{2}x(x+1)

\(\therefore\) LCM = a^{2}x(x+1)

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