Question

Find the LCM of the polynomials $8x^2-128$ and $2x^2-16x+32$.

Answer 1

We have
$p\left(x\right)=8x^2-128=8(x^2-16)=8(x^2-4^2)=2^3(x+4)(x-4)$;
$q\left(x\right)=2x^2-16x+32=2(x^2-8x+16)=2(x^2-2(x\left)\right(4)+4^2)=2{(x-4)}^2$.
We observe

irreducible factors of $p\left(x\right)$ or $q\left(x\right)$	Highest exponent
Numeric $2$	$3$
Algebraic $(x+4)$	$1$
Algebraic $(x-4)$	$2$

Hence the LCM is $2^3(x+4){(x-4)}^2=8(x+4){(x-4)}^2$.