To solve the problem of finding the factored form of [tex]\( x^9 + 27 \)[/tex], we need to determine which among the given options correctly represents this expression in its factored form.
Given options are:
A. [tex]\( \left(x^3 - 3\right)\left(x^6 + 3x^3 + 9\right) \)[/tex]
B. [tex]\( \left(x^3 + 3\right)\left(x^6 - 3x^3 + 9\right) \)[/tex]
C. [tex]\( (x - 3)^3\left(x^6 + 3x^3 + 9\right) \)[/tex]
D. [tex]\( (x + 3)^3\left(x^6 - 3x^3 + 9\right) \)[/tex]
To verify which factored form is correct, let us first consider the general method for factoring expressions similar to [tex]\( x^9 + 27 \)[/tex].
Notice that [tex]\( x^9 + 27 \)[/tex] can be interpreted using sum of cubes formula:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Letting [tex]\( a = x^3 \)[/tex] and [tex]\( b = 3 \)[/tex], we have:
[tex]\[ (x^3)^3 + 3^3 = (x^3 + 3)((x^3)^2 - x^3 \cdot 3 + 3^2) \][/tex]
[tex]\[ = (x^3 + 3)(x^6 - 3x^3 + 9) \][/tex]
We should compare this result with the options given:
A. [tex]\( \left(x^3 - 3\right)\left(x^6 + 3x^3 + 9\right) \)[/tex]
B. [tex]\( \left(x^3 + 3\right)\left(x^6 - 3x^3 + 9\right) \)[/tex]
C. [tex]\( (x - 3)^3\left(x^6 + 3x^3 + 9\right) \)[/tex]
D. [tex]\( (x + 3)^3\left(x^6 - 3x^3 + 9\right) \)[/tex]
Comparing our derived factored form [tex]\( (x^3 + 3)(x^6 - 3x^3 + 9) \)[/tex] to the options:
- Option A is incorrect because it has [tex]\( x^3 - 3 \)[/tex] and the second term does not match.
- Option B is correct, it matches our derived form [tex]\( (x^3 + 3)(x^6 - 3x^3 + 9) \)[/tex].
- Option C and D have different forms and exponents that do not match our derived factor.
Thus, the correct answer is:
[tex]\[ \boxed{B} \left(x^3 + 3\right)\left(x^6 - 3x^3 + 9\right) \][/tex]
