Find the HCF and LCM of:

Question

Please generate an appealing image related to math education. The image should depict a classroom setting. Show a group of people of different descents, such as a Caucasian teacher standing at the chalkboard, a Middle-Eastern woman and a South Asian man sitting, engaged in a conversation. Display a series of complex mathematical equations scribbled on the chalkboard, graph papers scattered on the teacher's desk, and calculators in various students' hands. Let the ambient lighting be soft yet clear, along with some potted plants in the background. Please ensure, however, that no text is visible in the image.

Find the HCF and LCM of:

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

Answer 1

First exp=1+4x+4x^2-16x^4

=(1)^2+2.1.2x+(2x)^2-16x^4
=(1+2x)^2-(4x^2)^2
=(1+2x+4x^2) (1+2x-4x^2)
Second exp.=1+2x-8x^3-16x^4
=(1)^3-(2x)^3+2x-16x^4
=(1-2x)(1+2x+4x^2)+2x(1-8x^3)
=(1+2x)(1-2x)(1+2x+4x^2)

:HCF=(1+2x+4x^2)

Answer 2

Find the HCF and LCM of:

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

Answer 3

2x^2(x^-a^2);3x^3(x^3-a^3) find lcm

Answer 4

in 2nd exp after taking -8x^3 common there should only be (1+2x)

Answer 5

The HCF of the first two expressions is indeed:

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

$1 + 2x - 8x^3 - 16x^4 = (1 + 2x)(1 - 2x + 4x^2) - 8x^3$

Therefore, the HCF is $1 - 2x + 4x^2$.

Answer 6

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

is there a significance to the .. at the end of the first term?

if not, then

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

1 + 2x - 8x^3 - 16x^4
= (1+2x) - 8x^3(1 + 2x^2)

16x^4 + 4x^2 + 1
= (16x^4 + 8x^2 + 1) - 4x^2
= (4x^2 + 1)^2 - 4x^2
= (4x^2 + 1 + 2x)(4x^2 + 1 - 2x)
= (4x^2+2x+1)(4x^2-2x+1)

At the moment I can't see anything else to do with the first two.