Which of the following is a factor of 500x^3+108y^18 5x+3y^6 25x^2+15xy^6+9y^2 All of the above

To determine which of the given options is a factor of \( 500 x^{3} + 108 y^{13} \), we can start by factoring the expression.<br /><br />First, notice that both terms in the expression have a common factor. The greatest common divisor (GCD) of the coefficients 500 and 108 is 4. Therefore, we can factor out 4:<br /><br />\[ 500 x^{3} + 108 y^{13} = 4(125 x^{3} + 27 y^{13}) \]<br /><br />Next, observe that \( 125 x^{3} \) and \( 27 y^{13} \) are perfect cubes:<br /><br />- \( 125 x^{3} = (5x)^3 \)<br />- \( 27 y^{13} = (3y^{6})^3 \cdot y \)<br /><br />The expression inside the parentheses, \( 125 x^{3} + 27 y^{13} \), can be factored using the sum of cubes formula:<br /><br />\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]<br /><br />Here, \( a = 5x \) and \( b = 3y^{6} \). Applying the formula gives:<br /><br />\[ 125 x^{3} + 27 y^{13} = (5x + 3y^{6})(25x^{2} - 15xy^{6} + 9y^{12}) \]<br /><br />Therefore, the original expression can be factored as:<br /><br />\[ 500 x^{3} + 108 y^{13} = 4(5x + 3y^{6})(25x^{2} - 15xy^{6} + 9y^{12}) \]<br /><br />From this factorization, it is clear that \( 5x + 3y^{6} \) is indeed a factor of the expression. <br /><br />Thus, the correct answer is:<br /><br />\( 5 x+3 y^{6} \)