Question

select the correct answer.
what is the factored form of ( x^9 + 27 )?

a. ( (x^3 - 3)(x^6 + 3x^3 + 9) )

b. ( (x^3 + 3)(x^6 - 3x^3 + 9) )

c. ( (x - 3)^3(x^6 + 3x^3 + 9) )

d. ( (x + 3)^3(x^6 - 3x^3 + 9) )

Explanation:

Step1: Recall Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\).

Step2: Rewrite the Expression

Rewrite \(x^9+27\) as \((x^3)^3 + 3^3\), where \(a = x^3\) and \(b = 3\).

Step3: Apply the Formula

Using the sum of cubes formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\), substitute \(a = x^3\) and \(b = 3\):
\((x^3)^3+3^3=(x^3 + 3)((x^3)^2 - x^3\times3 + 3^2)=(x^3 + 3)(x^6 - 3x^3 + 9)\)

Answer:

B. \((x^3 + 3)(x^6 - 3x^3 + 9)\)