Question

Find the LCM of $12x^4+324x,36x^3+90x^2-54x$

Answer 1

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

Factorise $12x^4+324x$ as follows:

$12x^4+324x=12x(x^3+27)=12x(x^3+3^3)=12x\left[\right(x+3\left)\right(x^2-(3\times x)+3^2\left)\right]=(2\times 2\times 3)x(x+3)(x^2+9-3x)$ (using identity $a^3+b^3=(a+b)(a^2+b^2-ab)$

Now, factorise $36x^3+90x^2-54x$ as follows:

$36x^3+90x^2-54x=18x(2x^2+5x-3)=18x(2x^2+6x-x-3)=18x\left[2x\right(x+3)-1(x+3\left)\right]$

$=(2\times 3\times 3)x(2x-1)(x+3)$

Therefore, the least common multiple between the polynomials $12x^4+324x$and $36x^3+90x^2-54x$ is:

$LCM=2\times 2\times 3\times 3\times x\times (2x-1)\times (x+3)\times (x^2+9-3x)=36x(2x-1)\left[\right(x+3\left)\right(x^2+9-3x\left)\right]$
$=36x(2x-1)(x^3+27)$ (using identity $a^3+b^3=(a+b)(a^2+b^2-ab)$)

Hence, the $LCM$ is $36x(2x-1)(x^3+27)$.