Algebra , Factoring, difference of Cubes PLEASE HELP!!!!?

Question

Ok I have to factor these some how n they are difference of cubes problems, I have not the slightest clue on what to do ... When I see these problems I go blank, I'm Having a test on a bunch of difference of cubes problems so pleas explain to me what the hell to do ...


Sum of Cubes Pattern
a3 + b3 = (a + b)(a2 - ab + b2)
Difference of Cubes Pattern
a3 - b3 = (a - b)(a2 + ab + b2)


1. 64x^3 + 1
2. 1-125b^3
3. 8x^3y^6 + 27
4. a^3 b^6 – b^3
5. 27x^3 – 8y^3

The most I know is you have to find the GCF first.....
Thanks........

Answer 1

I'll show you all the steps for the first one or two but after that it is straightforward. The hardest part is identifying the cubes. Simply take each part of the term in order and find the cube root.

PROBLEM 1:
64 x^3
--> 64 = 4^3
--> x^3 = x^3
= (4x)^3

So you can rewrite it as:
(4x)^3 + (1)^3

This becomes:
(4x + 1)[ (4x)² - (4x)(1) + (1)² ]

Then simplify:
= (4x + 1)(16x² - 4x + 1)

PROBLEM 2:
(1)^3 - (5b)^3
(1 - 5b)(1² + 5b + (5b)²)
(1 - 5b)(1² + 5b + 25b²)

PROBLEM 3:
8 x^3 y^6:
8 --> 2^3
x^3 --> x ^3
y^6 --> (y²)^3
= (2xy²)^3

(2xy²)^3 + (3)^3
You take it from here... first cube is 2xy², second cube is 3.

PROBLEM 4:
(ab²)^3 - (b)^3

PROBLEM 5:
(3x)^3 - (2y)^3

Answer 2

First memorize the patterns, then in each example rewrite the problem into the form of the pattern it corresponds to:
64x^3 + 1 = (4x)^3 + 1^3 = (4x + 1)(16x^2 - 4x + 1)
1 - 125b^3 = 1^3 - (5b)^3 = (1 - 5b)(1 + 5b + 25b^2)
The answers for the last three will be:
(2xy+3)(4x^2y^2 - 6xy + 9)
b^3(ab - 1)(a^2b^2 + ab +1), because you must first factor out the monomial b^3
(3x-2y)(9x^2 + 6xy +4y^2)
Now cover the answers and try to reconstruct the thought processes until you understand. Best wishes.

Answer 3

it's just all about using the patterns you have already typed above (the sum or the difference of cubes):

1. (4x + 1) (16x^2 - 4x + 1)
2. - (5b - 1) (25b^2 + 5b + 1)
3. (2xy^2 + 3) (4x^2y^4 - 6xy^2 + 9)
4. b^3 (ab - 1) (a^2b^2 + ab + 1)
5. (3x - 2y) (9x^2 +6xy + 4y^2)

Answer 4

1.) 64x³ + 1
= 4³x³ + 1
= (4x)³ + 1
= (4x + 1)((4x)²- 4x +1)
= (4x + 1)(16x² - 4x + 1)

2.) 1-125b³
= 1 - 5³b³
= (1 - 5b)(1 + 5b + 5²b²)
= (1 - 5b)(1 +5b + 25b²)

Answer 5

so, for number one think of it like this:

(4x)^3+(1)^3

then a=8x and b=1 and just plug into the formula:

[4x+1][(4x)^2-(4x)(1)+(1)^2]

now just simplify to get:

(4x+1)(64x^2-4x+1)

the others are the same really, the trick is just identifing the a and b of each problem.