Question

Find the LCM of $a^2-3a+2,a^3-a^2-4a+4,a(a^3-8)$

Answer 1

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

Factorise $a^2-3a+2$ as follows:

$a^2-3a+2=a^2-2a-a+2=a(a-2)-1(a-2)=(a-1)(a-2)$

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

$a^3-a^2-4a+4=a^2(a-1)-4(a-1)=(a^2-4)(a-1)=(a^2-2^2)(a-1)=(a+2)(a-2)(a-1)$

(using identity $a^2-b^2=(a+b)(a-b)$)

Finally, factorise $a(a^3-8)$ as follows:

$a(a^3-8)=a(a^3-2^3)=a(a-2)(a^2+2a+2^2)=a(a-2)(a^2+2a+4)$

(using identity $a^3-b^3=(a-b)(a^2+b^2+ab)$)

Therefore, the least common multiple between the polynomials $a^2-3a+2$, $a^3-a^2-4a+4$and $a(a^3-8)$ is:

$LCM=a\times (a-1)\times (a-2)\times (a+2)\times (a^2+2a+4)=a(a-1)(a+2)\left[\right(a-2\left)\right(a^2+2a+4\left)\right]$

$=a(a-1)(a+2)(a^3-8)$ (using identity $a^3-b^3=(a-b)(a^2+b^2+ab)$)

Hence, the $LCM$ is$a(a-1)(a+2)(a^3-8)$.