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HCF AND LCM OF POLYNOMIALS math capsule

I hate long articles or blog posts , don't you ?
That's why i write mine really very short, descriptive and covering all the concepts in one go .
This post is regarding the hcf and lcm of polynomials and monomials. If you want to learn about the hcf and lcm of numbers then click this yellow link below . It will take you to another article . Sorry ;)


 HCF and LCM 

HCF OF POLYNOMIALS                                     

The highest common factor of two or more algebraic expressions is the factor of the highest degree which divides each of the expressions without a remainder.
It is denoted by HCF or GCD.
It is a convention to take HCF in such a way that the coefficient of the highest power of the variable is positive.


HCF OF MONOMIALS

To find the HCF of two or more monomials , we multiply the HCF of the numerical coefficients of the monomials by the highest power of each of the letters common to both the polynomials.

The HCF of 48 z⁴ and 80 z⁵ is .....

So, HCF of 48 and 80 is 16 and HCF of z⁴ and z⁵ is z⁴.
Hence the required HCF will be 16 z⁴.

The HCF of x³ y² z⁵ and x² y⁴ z³ is ......

HCF of x³ and x² is x² , HCF of y² nd y⁴ is y² and HCF of z⁵ and z³ is z³.
Hence the required HCF of the given monomials is x² y² z³.


Great ! just a few concepts remaining .

HCF OF POLYNOMIALS

CASE 1 Which can be easily Factorised

STEP 1
Resolve both the polynomials in the complete factor form.
STEP 2
Required HCF is the product of the HCF of numerical coefficients of the polynomial and factors with their highest powers common to the polynomials.

The HCF of 22x(x + 1)² and 36x²(2x² +3x +1) is ......
HCF of 22 and 36 is 2.
Now, x(x + 1)² = x (x + 1)(x + 1)
x²(2x² + 3x + 1)= x²(2x + 1)(x + 1)
common factors of x(x+1)² and x²(2x² + 3x + 1) are x (x + 1) 
Hence required HCF = 2(x + 1)x

CASE 2 Which cannot be easily Factorised(successive division method)

STEP 1
Find the HCF of the monomial factors, if any of the given polynomials 
STEP 2
arrange the polynomials in ascending or descending order.
STEP 3
Divide the polynomial of higher degree by the polynomial of lower degree.
or If both the polynomials are of the same degree any one of them can be taken as divisor or dividend.
STEP 4
After the first division take the remainder as the new divisor and first divisor as new dividend.
STEP 5
Continue this process of dividing the last divisor by the last remainder until the remainder becomes zero.
STEP 6
Last divisor is the HCF of the polynomial.

POINTS TO BE REMEMBERED
1. If the first term of a remainder is negative at any stage, the sign of all of its term must be changed.
2. If at any stage, the remainder contains common factor take it out.
3. If the quotient exists as a fraction at any stage, then multiply the dividend by a suitable number to avoid fractional quotients.

 The HCF of 22x⁶ -78x⁵- 16x² and 2x⁵-78x²- 44x is.......
Here ,
22x⁶-78x⁵-16x² = 2x²(11 x⁴ -39x³ - 8)
2x⁵- 78x²-44x =2x (x⁴ - 39x -22)

to find HCF of 11x⁴-39x³ -8 and x⁴ -39x-22
x⁴-39x-22) 11x⁴-39x³ -8 (11
                   11x⁴-242-429x
                   -       +     +       
                   -39x³ +429x +234
-39 is taken out as common factor from remainder
x³-11x-6 ) x⁴-39x -22(x
                x⁴-11x²-6x
                -      +    +    
                11x² -33x -22

Again 11 is taken out as common factor from remainder
x²-3x-2) x³-11x-6 ( x+3
             x³- 3x² -2x
            -     +      +   
            3x²-9x -6
            3x²-9x -6
            -     +    +   
                 0

Therefore  HCF is x²-3x-2
also the hcf of 2x² and 2x is 2x.
required HCF = 2x( x² -3x -2) 

Read about HCF of polynomials ? Great , you are a fast learner .

Tips: make a separate notebook and take a few notes to remember the concepts easy!

LEAST COMMON MULTIPLE( LCM )

The lowest common multiple (LCM) of tow or more algebraical expressions is the expression of the lowest dimensions which is divisible by each of them without remainder.

LCM OF MONOMIALS

To find the LCM of two monomials , we multiply the LCM of the numerical coefficient of the monomials by all the factors raised to the highest power which it has in either of the given polynomials.

The LCM of 12x³y³z² and 18x⁴y²z³ is......
12x²y³z² = 2²* 3*x²*y³*z²
18x⁴y²z³ = 2 * 3²* x⁴*y²*z³
Required LCM = 2² * 3²* x⁴* y³ * z³
                         = 36 x⁴y³z³

LCM OF POLYNOMIALS

(which can be easily factorised)

We resolve both the polynomials as the product of different factors . Required LCM is the product of the LCM of numerical coefficients of both the polynomials and factors with their highest power which it has in either of the given polynomials.

POINTS TO BE REMEMBERED

> LCM of two polynomials = Product of two polynomials / HCF of two polynomials 

> HCF of two polynomials = Products of polynomials / LCM of polynomials 

> For any two polynomials p(x) and q(x) ,
     p(x) * q(x) = their HCF * their LCM


Huh ! we are done with the concepts . Now its time to do some exercises .

  
hcf and lcm of polynomials 1
HCF AND LCM OF POLYNOMIALS 1

hcf and lcm of polynomials 2
HCF AND LCM OF POLYNOMIALS 2

hcf and lcm of polynomials 3
HCF AND LCM OF POLYNOMIALS 3

hcf and lcm of polynomials 4
HCF AND LCM OF POLYNOMIALS 4


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