How do i factor a binomial when one term is cubed?

Question

I am being asked to "Factor Completely" this question:  27x³-1

Is this possible?

 

Answer 1

Yes it is possible.

 

Actually both of the terms in the equation are cubed.

 

27x³ = (3x)³  and (-1) = (-1)³= (-1)(-1)(-1)  = 1(-1) = -1  or (1)³ = 1

 

realizing this we can use formula for (a³ - b³) = (a-b)(a² + ab + b²)

in your example

• a = 3x
• b = 1

substituting these values in

 

(3x)³ - (1)³ = (3x - 1)((3x)² + (3x)(1) + 1²) = (3x -1) (9x² + 3x + 1)

 

9x² + 3x + 1 can't be factored any more so your answer is (3x -1)(9x² + 3x + 1)

 

to validate this answer do this by multiply out the answer

 

9x² + 3x + 1

*        3x - 1

---------------

27x³ + 9x² + 3x

+       (-9x² - 3x - 1)

--------------------------

27x³ + 0x² + 0x -1    ⇒   27x³ -1

 

Hope this helps

 

Answer 2

Hi Lizzie;

You are obviously working on the premise that only x² can be factored.  This is incorrect.  A factor can be...

 

(x²      )(x     )

or x³, x⁴, etc.

Henceforth

27x³-1

is factorable.

 

To factor, we must do the following...

 

We know that only

1x1=1

 

Henceforth...

(x²   1)(x   1)

 

We also know that one of the 1s must be negative to yield -1.  Therefore, one of the parenthetical equations must engage in subtraction, while the other addition.

 

To yield 27...

9 x 3

is the only possibility, and both numbers must either be negative or positive.

 

We also know that the FOIL equation must yield 0x.  The above will produce 1x² and 1x.

 

So we must start again as follows...

(x    1)(x   1)

is the only equation which will work.  However, we cannot forget about the third x.  Therefore...

x(x  + 1)(x -  1)

We also cannot forget about the 27.  Furthermore, both parenthetical x-s must be multiplied by the same number to yield 0x.

3x3x3 is the only mathematical formula which will yield 27.

 

3x(3x+1)(3x-1)

 

FOIL

FIRST...3x(9x³)

OUTER...3x(-3x)

INNER...3x(3x)

LAST...-1

 

27x³-9x²+9x²-1=27x³-1

 

Please let me know if you need anything else.

I LOVE MATH!!!!