Questions About How To Factor Cubes

Donald asks…

How do you factor cubes?

I've already read my textbook and also asked my teacher and I still don't understand it. I took a test and I failed the questions about factoring cubes. One of the questions is 3x^7-18x^5-21x^3. Can somebody explain to me how you would factor this? I really want to learn how to factor cubes.
Umm.. I don't think "OK" is the answer or factor of the problem. Please delete that if you can. Thanks.

admin answers:

First you can pull 3x^3 out, as you can clearly see it is a factor of each term.

3x^3(x^4 - 6x^2 - 7)

Now, only the product of a negative and positive is negative, so the middle signs of the binomial factors of x^4 - 6x^2 - 7 must be positive and negative.

3x^3(x^2 + 1)(x^2 - 7)

Above, 7 is prime, so its only factors are 1 and 7. Your job is to find factors who product is -7 (±1 and ±7) and whose sum is -6. You see my result above--1 & -7.

Multiply half a dozen binomials of your own design together, mix them up, and factor them. The experience should prove particularly useful to you.

For factoring cubes, synthetic division is handy. Try looking that up on google for pointers.

[Addendum: Another responder suggested x^2 + 1 could be factored. It cannot be factored as the difference of squares, as he suggests, however.]

Robert asks…

Factor each difference of cubes and factor each sum of cubes.?

i dont understand how to factor cubes

factor each difference of cubes
i) x3 - 1
x3 - 64

factor of each sum of cubes
x3 + 1
x3 +64

admin answers:

There are rules for that. You need to memorized them:
(a^3 + b^3) = (a + b)(a^2 - ab + b^2)
(a^3 - b^3) = (a + b)(a^2 + ab + b^2)
So using those rules:

x3 - 1 = (x - 1)(x^2 + x + 1)

x3 - 64 = (x - 4) (x^2 + 4x + 16)

x3 + 1 = (x + 1)(x^2 - 2x + 1)

x3 +64 = (x + 4)(x^2 - 4x + 16)

Steven asks…

how do you factor cubes?

k so i know how to factor squares those are super easy
but like cubes i know there's different steps

im doing the mental math portion of my exam tomorrow and i'm a bit iffy on this
its only outta 30 marks but i know there's going to be a difference of cubes question
i never knew how to do them and i dont want to lose this mark

so heres an example of a question that was on my last mental math (which i didnt even attempt)
(y^3 + x^3) + z^3 <----SIMPLIFY

the answer turned out to be something quite big thats all i can remember
can someone pleeeease explain to me whats going on here

please and thanks

admin answers:

Knowing the formula :

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

===========

(y^3 + x^3) = (y - x)(y^2 + xy + x^2) <====== calling it a & z = b

[ (y - x)(y^2 + xy + x^2) + z^3 ]

[ ((y - x)(y^2 + xy + x^2))^(1/3) - z ] * [ (((y - x)(y^2 + xy + x^2))^(1/3))^2 + ((y - x)(y^2 + xy + x^2))^(1/3) * z + (z)^2 ]

[ ((y - x)(y^2 + xy + x^2))^(1/3) - z ] * [ ((y - x)(y^2 + xy + x^2))^(2/3) + ((y - x)(y^2 + xy + x^2))^(1/3) * z + z^2 ]

Lisa asks…

hey how do I factor cubes?

here it is- (X+1)^3
factor and explain-0 then PLEASE give me the general formula (a+b)^3
which expands to somethng like (a+b)(a^2+something...) thanks guys

admin answers:

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

Carol asks…

how to factor cubes ?

216u^3 - v^3

so far i have (6u)^3 -(1v)^3
(2u- 1v)(3u^2 ...
and im stuckkk
please help. thanks so much

admin answers:

That is a difference of two cubes, and you have to use the formula.
(a - b)(a^2 +ab + b^2) for difference

So that gives you
(6u -v)(36u^2 +6uv - v^2)

For some of two cubes, your formula is (a + b)(a^2 - ab + b^2)

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