Answer:
LCM = [tex]8y^{4}(y+ 10)^{2}(y + 8)[/tex]
Step-by-step explanation:
Making factors of [tex]8y^{6}+ 144y^{5}+ 640y^{4}[/tex]
Taking [tex]8y^{4}[/tex] common:
[tex]\Rightarrow 8y^{4} (y^{2}+ 18y+ 80)[/tex]
Using factorization method:
[tex]\Rightarrow 8y^{4} (y^{2}+ 10y + 8y + 80)\\\Rightarrow 8y^{4} (y (y+ 10) + 8(y + 10))\\\Rightarrow 8y^{4} (y+ 10)(y + 8))\\\Rightarrow \underline{2y^{2}} \times 4y^{2} \underline{(y+ 10)}(y + 8)) ..... (1)[/tex]
Now, Making factors of [tex]2y^{4} + 40y^{3} + 200y^{2}[/tex]
Taking [tex]2y^{2}[/tex] common:
[tex]\Rightarrow 2y^{2} (y^{2}+ 20y+ 100)[/tex]
Using factorization method:
[tex]\Rightarrow 2y^{2} (y^{2}+ 10y+ 10y+ 100)\\\Rightarrow 2y^{2} (y (y+ 10) + 10(y + 10))\\\Rightarrow \underline {2y^{2} (y+ 10)}(y + 10) ............ (2)[/tex]
The underlined parts show the Highest Common Factor(HCF).
i.e. HCF is [tex]2y^{2} (y+ 10)[/tex].
We know the relation between LCM, HCF of the two numbers 'p' , 'q' and the numbers themselves as:
[tex]HCF \times LCM = p \times q[/tex]
Using equations (1) and (2): [tex]\Rightarrow 2y^{2} (y+ 10) \times LCM = 2y^{2} \times 4y^{2}(y+ 10)(y + 8) \times 2y^{2} (y+ 10)(y + 10)\\\Rightarrow LCM = 2y^{2} \times 4y^{2}(y+ 10)(y + 8) \times (y + 10)\\\Rightarrow LCM = 8y^{4}(y+ 10)^{2}(y + 8)[/tex]
Hence, LCM = [tex]8y^{4}(y+ 10)^{2}(y + 8)[/tex]
