SOLUTION: I am having trouble understanding how my instructor got this answer from factoring this problem. 125c^6 - 8d^6 =a^3-b^3= (a-b)*(a^2+ab+b^2) Can you please explain what he d

Algebra ->  Test -> SOLUTION: I am having trouble understanding how my instructor got this answer from factoring this problem. 125c^6 - 8d^6 =a^3-b^3= (a-b)*(a^2+ab+b^2) Can you please explain what he d      Log On


   

Question 688801: I am having trouble understanding how my instructor got this answer from
factoring this problem.
125c^6 - 8d^6
=a^3-b^3= (a-b)*(a^2+ab+b^2)
Can you please explain what he did?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 is not the answer.
It's more like the procedure to get the answer.
It's a formula affectionately called "difference of two cubes".
a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 or %28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3
is one of those "special products"
teachers and textbooks list.
Those special products "formulas" are true if you substitute
any number or expression for a
and any number or expression for b.
In the case of 125c%5E6+-+8d%5E6,
you are expected to notice that
125c%5E6=5%5E3%2A%28c%5E2%29%5E3=%285c%5E2%29%5E3, so your a is 5c%5E2
You are also expected to notice that
8d%5E6=2%5E3%2A%28d%5E2%29%5E3=%282d%5E2%29%5E3, so your b is 2d%5E2
Substituting your expressions for a and b into
a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 you get

You can go a little further, working on the right hand side and write it as
%285c%5E2-2d%5E2%29%2A%2825c%5E4%2B10c%5E2d%5E2%2B4d%5E4%29
A little more can be done, but it gets ugly.

NOTE:
%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3 is not a magical formula.
You can easily prove it if you multiply %28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29


EXTRA TIP:
Other special products are:
a%5E3%2Bb%5E3=%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29 or %28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29=a%5E3%2Bb%5E3 (sum of two cubes)
a%5E2-b%5E2=%28a%2Bb%29%28a-b%29 or %28a%2Bb%29%28a-b%29=a%5E2-b%5E2 difference of two squares
and also
%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2 (square of a binomial)

RANT:
This part of math (mulltiplying and factoring polynomials) is a nightmare for most students.
Unfortunately, factoring polynomials will keep coming back at various points through your current course and all the math courses that follow.
(Rational functions is one example).
It is useful to remember the special products formulas.
It is easier to remember them if you understand what they mean.
(The cute, affectionate names help).
And lots of practice with polynomials helps.
I was never a fan of homework, and rarely needed to do much practice with math, but I found out the hard way that this part of math required work.