Question

factor completely. 8u^6 + 27g^3 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. 8u^6 + 27g^3 = (factor completely. simplify your answer.) b. the polynomial is prime.

Explanation:

Step1: Recognize sum - of - cubes form

We know that \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\). Here, \(8u^{6}+27g^{3}=(2u^{2})^{3}+(3g)^{3}\), where \(a = 2u^{2}\) and \(b=3g\).

Step2: Apply sum - of - cubes formula

Substitute \(a = 2u^{2}\) and \(b = 3g\) into the formula \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\). We get \((2u^{2}+3g)[(2u^{2})^{2}-(2u^{2})(3g)+(3g)^{2}]\).

Step3: Simplify the second factor

\((2u^{2})^{2}=4u^{4}\), \((2u^{2})(3g)=6u^{2}g\), and \((3g)^{2}=9g^{2}\). So the factored form is \((2u^{2}+3g)(4u^{4}-6u^{2}g + 9g^{2})\).

Answer:

\(A. (2u^{2}+3g)(4u^{4}-6u^{2}g + 9g^{2})\)