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Chapter 0: Problem 62

Factor using the formula for the sum or difference of two cubes $$27 x^{3}-1$$

Short Answer

Expert verified
The factored form of the expression \(27x^3 - 1\) is \((3x - 1)(9x^2 + 3x + 1)\).

Step by step solution

01

Identify 'a' and 'b'

In the given expression, \(27x^3\) can be rewritten as \((3x)^3\) and 1 as \(1^3\). Therefore, 'a' is 3x and 'b' is 1.
02

Apply the formula for the difference of two cubes

The difference of two cubes can be factored as: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Substituting 'a' as 3x and 'b' as 1 into the formula, the expression can be factored as: (3x - 1)((3x)^2 + (3x)*1 + (1)^2).
03

Simplify the result

Simplify the resulting expression to get the final answer. This gives: (3x - 1)(9x^2 + 3x + 1).

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