Questions: Factor Completely: 8x^6 - 27y^6

Transcript text: Factor Completely: $8 x^{6}-27 y^{6}$

Solution

Solution Steps

To factor the expression $8x^6 - 27y^6$, recognize it as a difference of cubes. The expression can be rewritten as $(2x^2)^3 - (3y^2)^3$. Use the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Step 1: Identify the Expression

We start with the expression $8x^6 - 27y^6$. This can be recognized as a difference of cubes since it can be rewritten as $(2x^2)^3 - (3y^2)^3$.

Step 2: Apply the Difference of Cubes Formula

Using the difference of cubes formula, which states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, we set $a = 2x^2$ and $b = 3y^2$. Thus, we can factor the expression as follows:

\[ 8x^6 - 27y^6 = (2x^2 - 3y^2)((2x^2)^2 + (2x^2)(3y^2) + (3y^2)^2) \]

Step 3: Simplify the Second Factor

Now, we simplify the second factor:

\[ (2x^2)^2 = 4x^4, \quad (2x^2)(3y^2) = 6x^2y^2, \quad (3y^2)^2 = 9y^4 \]

Putting it all together, we have:

\[ (2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4) \]