Problem 57 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The factorized form of \(x^3 + 27\) using the formula for the sum of two cubes is \((x + 3)(x^2 - 3x + 9)\).

Step 1: Identify a and b in the problem

Taking \(x^3 + 27\) as \(a^3 + b^3\), we can identify \(a = x\) and \(b = 3\).

Step 2: Substitute a and b into the formula

Now we need to substitute \(a = x\) and \(b = 3\) into the formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). So after substitution, the formula becomes \((x + 3)(x^2 - 3x + 9)\).

Step 3: Final Answer

Thus, the factorized form of \(x^3 + 27\) using the formula for the sum of two cubes is \((x + 3)(x^2 - 3x + 9)\).