Factor the following polynomial completely showing all steps:

Question

27x^3+512

Answer 1

Hi Kerry.

 

This is a sum of cubes.   There is a formula that says

 

a³ + b³ = (a + b)(a² - ab + b²)

 

I make my students memorize this because it is the easiest way.

 

If you want to do long division with the (a + b) factor that is an option.  But memorizing the formula is the best.

 

We have

 

(3x)³ + (8)³  which correspond to a³  and b³  in the formula

 

This means that a = 3x   and   b = 8, now just put a=3x and b=8 everywhere they belong in the formula

 

 

(3x)³ + (8)³ = (3x + 8)[(3x)² - (3x)(8) + (8)²]

 

So factored completely:

 

(3x)³ + (8)³ = (3x + 8)(9x² - 24x + 64)

 

A few things to note:   the last term in the quadratic (64 here) is ALWAYS POSITIVE (added, never subtracted)

The other two signs (3x+8)  and (9x² - 24x ...)   must be opposite.  

 

Hope that helps.