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Chapter 0: Problem 60

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

Short Answer

Expert verified
The factored form of the expression \(x^3 - 27\) using the formula for the difference of two cubes is \((x - 3)(x^2 + 3x + 9)\).

Step by step solution

01

Identify the Cubes

The first step is to identify the cubes in our formula. Here, \(x^3\) is our first cube (\(a^3\)) and \(27\) is our second cube (\(b^3\)). Now we need to figure out the cube roots, which are \(x\) for \(x^3\) and \(3\) for \(27\). Now we have our \(a\) and \(b\), which are \(x\) and \(3\) respectively.
02

Apply the Formula

Now we will apply the formula for the difference of two cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Replacing \(a\) with \(x\) and \(b\) with \(3\), we get: \(x^3 - 3^3 = (x - 3)(x^2 + 3x + (3)^2)\).
03

Simplify the Result

The final step is to simplify the expanded equation: \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\).

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. A. Find \(\sqrt{16} \cdot \sqrt{4}\) B. Find \(\sqrt{16 \cdot 4}\) C. Based on your answers to parts (a) and (b), what can you conclude?

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Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(\frac{x^{-\frac{5}{4}} y^{\frac{1}{3}}}{x^{-\frac{3}{4}}}\right)^{-6} $$

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