snowlovelydayM

2021-09-11

Find the LCM of the given polynomial.
$3{x}^{2}-27,2{x}^{2}-x-15$

The lowest common multiple of 2 polynomials can be obtained factorizing both the polynomials. The factors that are common are taken only once in the product term of the lowest common multiple.
The factors that are not common are then multiplied with the common factors to obtain the lowest common multiple. The lowest common multiple of polynomials is also a polynomial.
The polynomial $3{x}^{2}-27$ is factorized as follows:
$3{x}^{2}-27=3\left({x}^{2}-9\right)$
$=3\left(x+3\right)\left(x-3\right)$
The polynomial $2{x}^{2}-x-15$ is factorized as follows:
$2{x}^{2}-x-15=2{x}^{2}-6x+5x-15$
$=2x\left(x-3\right)+5\left(x-3\right)$
$=\left(2x+5\right)\left(x-3\right)$
The common factor between the 2 polynomials is $\left(x-3\right)$. The other factors are obtained as 3, $\left(x+3\right)$ and $\left(2x+5\right)$. Thus the lowest common multiple is the product of these factors and obtained as follows:
$LCM=3\left(x-3\right)\left(x+3\right)\left(2x+5\right)$
$=3\left({x}^{2}-9\right)\left(2x+5\right)$
$=\left(3{x}^{2}-27\right)\left(2x+5\right)$
$=6{x}^{3}+15{x}^{2}-54x-135$
Thus the lowest common multiple of $3{x}^{2}-27$ and $2{x}^{2}-x-15$ is obtained to be $6{x}^{3}+15{x}^{2}-54x-135$

Do you have a similar question?