Find the HCF and LCM of:

1 + 4x + 4x^2 - 16x^4.. ,1 + 2x - 8x^3 - 16x^4.., 16x^4 + 4x^2 + 1

what's with the .. ?

nothing. that is just to separate the equations.. please help

hmmm. None of the polynomials factors over the reals, so I don't see any help there.

Looks like GCF=1 and
LCM = (1+4x+4x^2-16x^4)(1+2x-8x^3-16x^4)(16x^4+4x^2+1)

if
x=-1 we have -17,-7,21
x=0 we have 1,1,1
x=1 we have -7,-21,21
x=2 we have -231,-315,273

To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the given expressions, we will factorize each expression first.

Expression 1: 1 + 4x + 4x^2 - 16x^4
Rearranging and grouping the terms, we get:
(1 - 16x^4) + (4x + 4x^2)
Taking out the common factor from the first two terms, we have:
-15x^4 + 1 + 4x + 4x^2
Factoring this further, we get:
(1 - 15x^4) + (4x + 4x^2)
Taking out the common factor from the second pair of terms, we have:
(1 - 15x^4) + 4x(1 + x)
Factoring the first term using the difference of squares, we have:
(1 - 4x^2)(1 + 4x^2) + 4x(1 + x)

Expression 2: 1 + 2x - 8x^3 - 16x^4
Rearranging and grouping the terms, we get:
(1 - 16x^4) + (2x - 8x^3)
Taking out the common factor from the first two terms, we have:
-15x^4 + 1 + 2x - 8x^3
Factoring this further, we get:
(1 - 15x^4) + (2x - 8x^3)
Taking out the common factor from the second pair of terms, we have:
(1 - 15x^4) + 2x(1 - 4x^2)

Expression 3: 16x^4 + 4x^2 + 1
This expression cannot be factorized further. It is already in its simplest form.

Now, let's find the HCF and LCM:

HCF:
To find the HCF, we need to identify the highest power of each common factor in the multiple expressions.
From the factorization of Expression 1 and Expression 2, we can see that the common factor is (1 - 15x^4).

Therefore, the HCF is (1 - 15x^4).

LCM:
To find the LCM, we multiply the highest power of each factor in the multiple expressions.

In Expression 1, the factors are (1 - 4x^2) and (1 + 4x^2).
In Expression 2, the factors are (1 - 4x^2) and (1 + 4x^2).
In Expression 3, the factors are (16x^4 + 4x^2 + 1).

Multiplying the highest power of each factor gives us the LCM:
(1 - 4x^2)(1 + 4x^2)(16x^4 + 4x^2 + 1)

Therefore, the LCM is (1 - 4x^2)(1 + 4x^2)(16x^4 + 4x^2 + 1).