Thread: Difference of Cubes

I have a real hard time understading the difference of cubes.
However, there are several questions on this topic in the coming June state exam.

How do I play with this topic?

QUESTIONS:

(1) Factor a^3b^3 - 8x^6y^9

(2) x^3 + 27

Originally Posted by symmetry

I have a real hard time understading the difference of cubes.
However, there are several questions on this topic in the coming June state exam.

How do I play with this topic?

QUESTIONS:

(1) Factor a^3b^3 - 8x^6y^9

(2) x^3 + 27

You have to memorize the factoring of the forms (x^3 +y^3) and (x^3 -y^3).
x^3 +y^3 = (x+y)(x^2 -xy +y^2) -----(i)
x^3 -y^3 = (x-y)(x^2 +xy +y^2) ------(ii)

(1) (a^3)(b^3) -8(x^6)(y^9)
= (ab)^3 -[2(x^2)(y^3)]^3
= [(ab) -2(x^2)(y^3)]*{(ab)^2 +(ab)[2(x^2)(y^3)] +[2(x^2)(y^3)]^2}
= [ab -2(x^2)(y^3)]*[(a^2)(b^2) +2ab(x^2)(y^3) +4(x^4)(y^6)] ----answer.

(2) x^3 +27
= x^3 +3^3
= (x+3)(x^2 -x*3x +3^2)
= (x+3)(x^2 -3x +9) -------------answer.

Originally Posted by symmetry

I have a real hard time understading the difference of cubes.
However, there are several questions on this topic in the coming June state exam.

How do I play with this topic?

QUESTIONS:

(1) Factor a^3b^3 - 8x^6y^9

(2) x^3 + 27

Ticbol says you have to memorise the factorisation of the sum and
difference of two cubes, but in fact if you know that sums or differences
of two cubes are involved you can deduce what the formulas are on the fly.

Consider f(x) = x^3+y^3, if x=-y, then f(x)=0, so (x+y) is a factor of f(x).

so we may write:

x^3+y^3=(x+y)(x^2 + A xy + y^2) = x^3 + A x^2y + xy^2 +yx^2 + Axy^2+y^3

Collect the terms with like powers of x and y:

x^3+y^3=(x+y)(x^2 + A xy + y^2) = x^3 + (A+1) x^2y + (A+1)yx^2 + y^3

So as the two sides are equal (A+1)=0, or A=-1, and:

x^3+y^3=(x+y)(x^2 - xy + y^2).

This could have been obtained using polynomial long division once we know
that (x+y) is a factor of f(x).

The other factorisation can be obtained in a similar manner by observing
that (x-y) is a factor of x^3-y^3, or by replacing y by -y' throughout
the factorisation already obtained.

RonL

Hello, symmetry!

I assume you know the sum and difference formulas.
. .

Where is your difficulty?
. . Understanding them?
. . Memorizing them?

. . . . . . . . . . . . . . . . . . . . . . .
First, we must recognize the form: .
. . . . . . . . . . . . . . . . . . . . . . . .


The cubes break up into a linear factor and a quadratic factor.

. . . . . .


To memorize the signs, think of the word SOAP.

. .
. . . . . . . . . . . . . .
. . . . . . . . . . . . Same . . Opposite . Always Positive

Actually you don't need to memorize them, though memorization IS handy. As long as you recall that is divisible by I think this provides a good exercise for that long division you mentioned in another thread.

-Dan

Originally Posted by topsquark

Actually you don't need to memorize them, though memorization IS handy. As long as you recall that is divisible by I think this provides a good exercise for that long division you mentioned in another thread.

-Dan

There seems to be a bit of an echo in here today .

RonL

I want to thank every tutor for your insight and tips.

Yes, my problem has been memorizing difference of cubes formulas.

Originally Posted by CaptainBlack

There seems to be a bit of an echo in here today .

RonL

Oops! I guess it pays to read the details of each post in the thread, which I obviously didn't do here.

-Dan