I really need help with factoring polynomials. I used to be fine with them, but I was just introduced to factoring cubes, and now I'm totally lost.

Could someone please show me step by step instruction on how I should factor these polynomials?

27x^3 + 8

y^3 - 216

y^3 +27

To factorise polynomial of cubes; two formulae apply
1) a^3+b^3
= (a+b)(a^2-ab+b^2)
2) a^3-b^3
=(a-b)(a^2+ab+b^2)

27X^3+8
=(3X)^3+2^3
Taking 3X as 'a' and 2 as 'b',and applying formulae 1.
=(3X+2)(9X^2-6X+4)

Y^3-216
=Y^3-6^3
Using formula 2
=(Y-6)(Y^2+6Y+36)

Y^3+27
=Y^3+3^3
=(Y+3)(Y^2-3Y+9)

hope this help.

@poster. First you need to know there formulas for difference of 2cubes and addition of 2cubes:
addition,
a^3+b^3=(a+b)(a^2-ab+b^2)
also for difference,
a^3-b^3=(a-b)(a^2+ab+b^2)
you can expand these for checks.
Now to your question.
1) 27x^3+8
you need to express in another form. You ought to know that the expression above is same as (3x)^3+(2)^3
hence using the formula for addition of 2cubes stated above,
27x^3+8=(3x)^3+(2)^3
(3x)^3+(2)^3=(3x+2)((3x)^2-3x*2+2^2)
finally We've
(3x)^3+(2)^3=(3x+2)(9x^2-6x+4).QED.
Same goes for number 3. Use additio of 2cubes and change 27 to 3^3.
Number 2.
Y^3-216.
This is difference of 2 cubes, where 216=6^3.
Therefore,
y^3-216=y^3-6^3
using the formula above for difference of 2cubes, we then have,
y^3-6^3=(y-3)(y^2+3y+3^2)
we finally obtain,
y^3-6^3=(y-3)(y^2+3y+9).
You have to note that these procedure is step-by-step and in each solution you have to state the formula you using be it difference or addition of 2cubes or both.

@ bollybees & funkystanl

Thank you both very much, I looked at both of your explanations and they have helped me tremendously!