I don't understand this (a+b)^3 thing, please help?

Question

1) is (a+b)^3 = (a+b)(a+b)(a+b)?

and if so wouldn't that be (a+b)(a^2 + 2ab + b^2)?

2) ex. prob... 8x^3 + 27

could i rewrite that as (2x + 3)^3?

and therefore set up the equation using the above as

(2x+3)(4x^2 + 12x + 9)?

but when i looked in the answers section it said that

8x^3 + 27 = (2x+3)(4x^2 - 6x + 0)

why is that so? can someone please explain

Answer 1

1)

(a + b)^3

= (a + b)(a + b)(a + b)

= (a*a + b*a + a*b + b*b)(a + b)

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

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

= a^2*a + 2ab*a + b^2*a + a^2*b + 2ab*b + b^2*b

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

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

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

2)

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

8x^3 + 27 = (2x)^3 + 3^3 = (2x + 3)(4x^2 - 6x + 9)

(2x + 3)(4x^2 - 6x + 0)

= (2x + 3)(4x^2 - 6x)

= 2x*4x^2 + 3*4x^2 - 2x*6x - 3*6x

= 8x^3 + 12x^2 - 12x^2 - 18x

= 8x^3 - 18x

∴ 8x^3 + 27 ≠ (2x + 3)(4x^2 - 6x + 0)

Answer 2

8x^3 + 27 is not (2x + 3)^3

its

(2x)^3 + 3^3

a^3 + b^3 is the sum of 2 cubes

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

Answer 3

Your 1) is correct but 2) is wrong

8x^3+27 does not equal (2x+3)^3

(2x+3)^3 = (2x+3)(2x+3)(2x+3)=

(2x+3)(4x^2 + 12x +9)

It is true 8x^3+27 = (2x+3)(4x^2-6x+9)

So you see they're not the same.

ANSWER