Question

Which expression is equivalent to 8x3 − 27 when factored completely?

Answer 1

The given expression is

`8x^3-27`

To find the equivalent expression, we use the formula for the difference of perfect cubes.

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

But, we have to express the numbers 8 and 27 as cubic powers.

`\begin{gathered} 8=2\cdot2\cdot2=2^3 \\ 27=3\cdot3\cdot3=3^3 \end{gathered}`

Then, we know that a and b are

`\begin{gathered} a=2x \\ b=3 \end{gathered}`

So, the difference would be

`(2x)^3-(3)^3`

Once we have the difference expressed in cubes, we apply the formula using the a and b values we determined before.

`(2x)^3-(3)^3=(2x-3)((2x)^2+(2x)(3)+(3)^2)_{}=(2x-3)(4x^2_{}+6x+9)`

Therefore, the factors are

`(2x-3)(4x^2+6x+9)_{}`