Question

Find the LCM of $6a^3+60a^2+150a,3a^4+12a^3-15a^2$

Answer 1

We know that $LCM$ is the least common multiple.

Factorise $6a^3+60a^2+150a$ as follows:

$6a^3+60a^2+150a=6a(a^2+10a+25)=6a[a^2+(2\times 5\times a)+5^2]=6a(a+5{)}^2$

(using identity $(a+b{)}^2=a^2+b^2+2ab$)

Now, factorise $3a^4+12a^3-15a^2$ as follows:

$3a^4+12a^3-15a^2=3a^2(a^2+4a-5)=3a^2(a^2+5a-a-5)=3a^2\left[a\right(a+5)-1(a+5\left)\right]=3a^2(a-1)(a+5)$

Therefore, the least common multiple between the polynomials $6a^3+60a^2+150a$and $3a^4+12a^3-15a^2$ is:

$LCM=6\times a^2\times (a+5{)}^2\times (a-1)=6a^2(a+5{)}^2(a-1)$

Hence, the $LCM$ is$6a^2(a+5{)}^2(a-1)$.