Question: Factor completely. \[ 243x^4 - 3y^4 \] \[ 243x^4 - 3y^4 = \boxed{\phantom{expression}} \]

Factor completely.

\[ 243x^4 - 3y^4 \]

\[ 243x^4 - 3y^4 = \boxed{\phantom{expression}} \]

Solution

To factor the expression completely, we have: \[ 243x^4 - 3y^4 \] First, factor out the greatest common factor. The greatest common factor is 3: \[ 243x^4 - 3y^4 = 3(81x^4 - y^4) \] Next, notice that \(81x^4 - y^4\) is a difference of squares: \[ 81x^4 - y^4 = (9x^2)^2 - (y^2)^2 \] Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can further factor: \[ (9x^2)^2 - (y^2)^2 = (9x^2 - y^2)(9x^2 + y^2) \] The expression can now be written as: \[ 243x^4 - 3y^4 = 3(9x^2 - y^2)(9x^2 + y^2) \] Finally, consider if \(9x^2 - y^2\) can be factored further: \[ 9x^2 - y^2 = (3x - y)(3x + y) \] So, the completely factored form is: \[ 243x^4 - 3y^4 = 3(3x - y)(3x + y)(9x^2 + y^2) \]