Question

Find the least common multiple of 25k^4+200k^3+400k^2 and 15k^3+90k^2+120k answer in its factored form

Answer 1

Answer:

$75k(x+4)(x+2)$

Step-by-step explanation:

The first expression is

$25k^4+200k^3+400k^2$

First, we extract the greater common factor

$25k^{2}(k^{2}+8k+16 )$

Second, we have the trinomial $k^{2}+8k+16$

Notice that we need to find two number which product is 16 and which sum is 8, those numbers are 4 and 4.

So, the factors of the expression are

$25k^4+200k^3+400k^2=25k^{2} (x+4)(x+4)=25k^{2}(x+4)^{2}$

The second expression is

$15k^3+90k^2+120k$

We use the same process.

Extract the greatest common factor

$15k(k^{2}+6k+8 )$

We solve the trinomial $k^{2}+6k+8$. We need to find two numbers which product is 8 and which sum is 6, those numbers are 4 and 2.

$15k^3+90k^2+120k=15k(x+2)(x+4)$

Now, the least common multiple is formed by factors that repeats in both expression.

Therefore, the least common multiple is $75k(x+4)(x+2)$, because the LCM between 25 and 15 is 75.