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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 factorization of \(x^{3}-27\) using the difference of two cubes is \((x - 3)(x^{2} + 3x + 9)\).

Step by step solution

01

Identify the Cubes

In the given expression \(x^{3}-27\), identify the two cubes. In this case, the two cubes are \(x^{3}\) and \(27\). Here, \(x^{3}\) is the cube of \(x\) and \(27\) is the cube of \(3\). So, we can rewrite the expression as \((x)^{3} - (3)^{3}\).
02

Apply the Formula

The formula for the difference of two cubes is \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). Apply this formula to the expression. Following the formula, \(a\) corresponds to \(x\) and \(b\) corresponds to \(3\). So, replace \(a\) with \(x\) and \(b\) with \(3\) in the formula. We get \((x - 3)((x)^{2} + x*3 + (3)^{2})\).
03

Simplify the Results

Simplify the result obtained in the previous step to have the final factorized form.Simplify the expression \((x)^{2} + x*3 + (3)^{2}\) to obtain \(x^{2} + 3x + 9\).So, the factorization of the expression \(x^{3}-27\) using the difference of two cubes is \((x - 3)(x^{2} + 3x + 9)\).

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When checking a radical equation's proposed solution, I can substitute into the original equation or any equation that is part of the solution process.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The rational expressions $$\frac{7}{14 x} \text { and } \frac{7}{14+x}$$ can both be simplified by dividing each numerator and each denominator by 7

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{3}-64=(x+4)\left(x^{2}+4 x-16\right)$$
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