Find the L.C.M of the following. a. \: 1 + 4x + 4 {x}^(2) - 16 {x}^4 \: and \: 1 + 8 {x}^(3) ​

Question

Find the L.C.M of the following.


`a. \: 1 + 4x + 4 {x}^(2) - 16 {x}^4 \: and \: 1 + 8 {x}^(3)`
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Answer 1

Explanation:

To find the L.C.M of the given polynomials, we need to factorize them first.

1 + 4x + 4{x}^2 - 16{x}^4 can be written as (1-2x{^2}){^2}.

1 + 8{x}^3 can be written as (1+2{x}^3).

So, the L.C.M would be the product of the highest powers of each factor.

In this case, (1-2x{^2}){^2} has the highest power of 2, while (1+2{x}^3) has the highest power of 1.

Thus, the L.C.M of 1 + 4x + 4{x}^2 - 16{x}^4 and 1 + 8{x}^3 is (1-2x{^2}){^2} (1+2{x}^3).

Answer 2

First, let's factor the two polynomials:

1 + 4x + 4x^2 - 16x^4 = (1 - 2x)^2(1 + 2x)

1 + 8x^3 = (1 + 2x)(1 - 2x + 4x^2)

To find the LCM of these two expressions, we need to take each factor to its highest power. Thus, the LCM is:

(1 - 2x)^2(1 + 2x)(1 + 2x^2)