Sure, let's go through the detailed steps to factor the expression [tex]\( 8x^3 + 27a^6 \)[/tex] and identify the correct answer.
Step 1: Identify the form of the given expression.
The expression [tex]\( 8x^3 + 27a^6 \)[/tex] can be recognized as a sum of cubes. In general, the sum of cubes is factored using the formula:
[tex]\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \][/tex]
Step 2: Identify [tex]\( A \)[/tex] and [tex]\( B \)[/tex] for the given expression.
Here,
[tex]\[ A = 2x \][/tex]
[tex]\[ B = 3a^2 \][/tex]
This is because:
[tex]\[ (2x)^3 = 8x^3 \][/tex]
[tex]\[ (3a^2)^3 = 27a^6 \][/tex]
Step 3: Apply the sum of cubes formula.
Using the formula [tex]\( A^3 + B^3 = (A + B)(A^2 - AB + B^2) \)[/tex]:
[tex]\[
8x^3 + 27a^6 = (2x + 3a^2)((2x)^2 - (2x)(3a^2) + (3a^2)^2)
\][/tex]
Step 4: Simplify inside the parentheses.
Calculate each term inside the second parentheses:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
[tex]\[
(2x)(3a^2) = 6xa^2
\][/tex]
[tex]\[
(3a^2)^2 = 9a^4
\][/tex]
Therefore:
[tex]\[
8x^3 + 27a^6 = (2x + 3a^2)(4x^2 - 6xa^2 + 9a^4)
\][/tex]
Step 5: Verify the options.
Comparing the factored form with the given options:
(A) [tex]\(\left(2x + 3a^2\right)^3\)[/tex]
(B) Prime
(C) [tex]\(\left(2x + 3a^2\right)\left(4x^2 - 6xa^2 + 9a^4\right)\)[/tex]
(D) None of these
The correct factorization aligns with Option (C):
[tex]\[
\left(2x + 3a^2\right)\left(4x^2 - 6xa^2 + 9a^4\right)
\][/tex]
Thus, the correct answer is:
(C) [tex]\(\left(2x + 3a^2\right)\left(4x^2 - 6xa^2 + 9a^4\right)\)[/tex]
The first digit of your code is 2.
