factoring completly?!?!?!?!?

Question

okay i need to factor this problem out completly, i can normally do these but this would be a sum of 2 cubes because theres only one cube... and i know this isnt a polynomial... so please help!
8x^3+125

fine maybe im wrong...but still how would you factor it? jeez i have a final on this in like and hour!!!!!

Answer 1

You do have a sum of cubes:
8x^3 + 125 = (2x)^3 + 5^3

To factor a sum of cubes:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)

here a = 2x, b = 5, so
8x^3 + 125 = (2x + 5)(4x^2 - 10x + 25)

Answer 2

There ARE two cubes: 125 = 5^3.

You seem to indicate that you would now know how to do it, so let me not spoonfeed you, as some will undoubtedly do.

The simplest way is to first re-express (y^3 + z^3) in the way you know how, and then just fill in from there. A most complete way of showing that you've mastered factoring would then be to show that the quadratic expression left for one of the factors CAN'T itself be factored, since, if set equal to zero its roots would be COMPLEX.

Good luck!

Live long and prosper.

POSTSCRIPT

Remember the old adage: "Give pipqueen a plug-in formula, and she can work one problem; give her a METHOD, and she can use it for life."

You'll notice that Jim Burnel..., and others, below, HAVE done the factoring for you, and therefore removed the opportunity for you to really learn anything for yourself.

You might ask a question that no-one else has addressed, so far (in the 6 answers below). How could you have told that (y + z) itself would be a factor? The answer is that y = - z clearly makes the sum of the cubes zero, so that (y + z) is zero. That means that (y + z) is a factor. Once you know that, you can just write down:

y^3 + z^3 = (y + z) (a y^2 + b yz + c z^2), and check out what a, b, and c must be.

But notice once more: in all my experience of seeing questions like this, no-one else --- NO SINGLE PERSON --- has ever (EVER!) given any REASON why the quadratic residue isn't factorizable in REAL terms. By not doing that, they are, in effect, saying "Trust me on this one," in other words, "Accept my parroted authority." That is both unmathematical and unscientific. So there.

(In answer to a message objecting to my name: I am Dr Spock, not Dr. Spock. Note the essential difference, though that is not pronounced. The reason why my own doctorate was ignored in your terrestrial television series has been spelled out in another Yahoo! Answer. It was that I should not be confused with Dr. Spock the American paediatrician, particularly as my role in the series was to child-sit humans as they took their first tentative baby steps into the far reaches of the Universe.)

Answer 3

There are 2 cubes...

(2x + 5)(4x^2 - 10x + 25)

In general:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

In this case a = 2x and b = 5

Answer 4

There are in fact 2 cubes here.

8x^3 = (2x)(2x)(2x) and
125 = 5*5*5

so then the factorization is
(2x+5)(4x^2-10x+25)

I agree that this is the answer.

Answer 5

You can do it. It's the sum of two cubes. 5^3=125.

Answer 6

There are in fact 2 cubes here.

8x^3 = (2x)(2x)(2x) and
125 = 5*5*5

so then the factorization is
(2x+5)(4x^2-10x+25)

Answer 7

8x^3+125
=(2x)^3+(5)^3
(2x+5)(4x^2-10x+25)

Answer 8

(8x)(8x)(8x) + 125
(64x^2)(8x) + 125
512x^3 + 125