B2 Find the greatest common divisor and the least common multiple of the expressions: $24 x^{4}-24 x^{3}+6 x^{2}$, $64 x^{3}-8,48 x^{2}-12$.
B2. We factorize the expressions separately: $24 x^{4}-24 x^{3}+6 x^{2}=6 x^{2}\left(4 x^{2}-4 x+1\right)=6 x^{2}(2 x-1)^{2}$,

$64 x^{3}-8=8(2 x-1)\left(4 x^{2}+2 x+1\right)$ and

$48 x^{2}-12=12\left(4 x^{2}-1\right)=12(2 x-1)(2 x+1)$. We determine that the greatest common divisor is $D=2(2 x-1)$ and the least common multiple is $v=24 x^{2}(2 x-1)^{2}(2 x+1)\left(4 x^{2}+2 x+1\right)$.

Factorization of the expression $24 x^{4}-24 x^{3}+6 x^{2}=6 x^{2}\left(4 x^{2}-4 x+1\right)=6 x^{2}(2 x-1)^{2} \ldots \ldots \ldots \ldots \ldots .1$ point

Factorization of the expression $64 x^{3}-8=8(2 x-1)\left(4 x^{2}+2 x+1\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-11.jpg?height=60&width=1685&top_left_y=1238&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-11.jpg?height=54&width=1685&top_left_y=1298&top_left_x=228)

Least common multiple $v=24 x^{2}(2 x-1)^{2}(2 x+1)\left(4 x^{2}+2 x+1\right) \ldots \ldots \ldots \ldots . . .1$ point