Factoring the sum or difference of two cubes

Question

I'm learning about sums and differences of cubes and I can't understand it very well. I am faced with this problem:

$$x^3 - 27$$

I am told to find the sum or difference of the two cubes. I understand I need to get it into the form of $a^3 - b^3$, which turns out to this:

$$x^3 - 3^3$$

I don't know how to find the sum or difference of cubes. If anyone would care to explain how to do this so I could solve this and other problems like it I would appreciate it. Thank you for your time.

Answer 1

(STRONG) HINT:If you multiply $(A+B)(A^2-AB+B^2)$ you get $A^3+B^3$ similarly $A^3-B^3=(A-B)(A^2+AB+B^2)$. So you have $A=x$ and $B=3$. Substitute into the above formula and you will get the factored form.

Answer 2

can you identify a pattern here? $$x^6+x^5y+x^4y^2+x^3y^3+x^2y^4+xy^5+y^6$$ $$x-y$$

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$$\begin{array}{rrrrrrrrrrr} x^7&+&x^6y&+&x^5y^2&+&\dots&+&xy^6&&\\ &-&x^6y&-&x^5y^2&-&\dots&-&xy^6&-&y^7\\ \end{array}$$

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$$\begin{array}{rrrrrrrrrrr} x^7&+&0x^6y&+&0x^5y^2&+&\dots&+&0xy^6&-&y^7 = x^7-y^7\\ \end{array}$$

Now try $$(x+y)(x^6-x^5y+x^4y^2-x^3y^3+x^2y^4-xy^5+y^6)$$