Question

Find the LCM of the following polynomials # a^8- b^8 and a^4- b^4×( a+b )

Answer 1

The LCM of the polynomial is  ( $a^{4}$ + $b^{4}$ ) (a² + b² ) ( a + b ) ( a - b )

Step-by-step explanation:

Given as :

The two polynomials are

$a^{8} - b^{8}$                      ........1

( $a^{4} - b^{4}$ ) (  a + b )               ..................2

To calculate the LCM of polynomials

According to question

Now, Expansion of polynomials

From polynomial 1

$a^{8} - b^{8}$  = $(a^{4})^{2}$  -  $(b^{4})^{2}$

            =[  $(a^{4})$ + $(b^{4})$  ] [ $(a^{4})$ - $(b^{4})$ ]

            = [ $(a^{4})$ + $(b^{4})$  ] [ $(a^{2})^{2}$ - $(b^{2})^{2}$]

            = [ $(a^{4})$ + $(b^{4})$  ] [ { a² + b² } { a² - b² } ]

            = [ $(a^{4})$ + $(b^{4})$  ] [ { a² + b² } { ( a + b ) ( a - b ) } ]

            = ( $(a^{4})$ + $(b^{4})$ ) ( a² + b² )  ( a + b ) ( a - b )

From polynomial 2

( $a^{4} - b^{4}$ ) (  a + b )  =  [ $(a^{2})^{2}$ - $(b^{2})^{2}$] ( a + b )

                              =   [ { a² + b² } { a² - b² } ] ( a + b )

                              = [ { a² + b² } { ( a + b ) ( a - b ) } ] ( a + b )

                              = ( a² + b² )  ( a + b ) ( a - b ) ( a + b )

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

Now, LCM of the polynomial

LCM of two polynomial is the expression of lowest degree exactly divisible by each of given expression

So, From the expansion of both the polynomial , we get

LCM of the polynomial = ( $a^{4}$ + $b^{4}$ ) (a² + b² ) ( a + b ) ( a - b )

Hence, The LCM of the polynomial is ( $a^{4}$ + $b^{4}$ ) (a² + b² ) ( a + b ) ( a - b ) Answer