Question

select the correct answer.
what is the factored form of ( 1458x^3 - 2 )?

a. ( 2(9x - 1)(81x^2 + 9x + 1) )
b. ( 2(9x + 1)(81x^2 - 9x + 1) )
c. ( (9x - 2)(81x^2 + 18x + 4) )
d. ( (9x + 2)(81x^2 - 18x + 4) )

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of \(1458x^3\) and \(2\). The GCF of \(1458\) and \(2\) is \(2\). So, factor out \(2\) from the expression:

$$1458x^3 - 2 = 2(729x^3 - 1)$$

Step2: Recognize the difference of cubes

Notice that \(729x^3=(9x)^3\) and \(1 = 1^3\). So, \(729x^3 - 1\) is a difference of cubes, which has the formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). Here, \(a = 9x\) and \(b = 1\).

Step3: Apply the difference of cubes formula

Applying the formula to \(729x^3 - 1=(9x)^3 - 1^3\):

$$
LATEXBLOCK0
$$

Step4: Combine with the factored GCF

Recall that we factored out \(2\) earlier, so putting it all together:

$$1458x^3 - 2 = 2(9x - 1)(81x^2 + 9x + 1)$$

Answer:

A. \(2(9x - 1)(81x^2 + 9x + 1)\)