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Chapter 0: Problem 63

Factor using the formula for the sum or difference of two cubes. $$64 x^{3}+27$$

Short Answer

Expert verified
The factorization of \(64x^{3}+27\) using the sum of cubes formula results in \((4x+3)(16x^{2}-12x+9)\)

Step by step solution

01

Express the Expression as a Sum of Cubes

In this case, \(64x^{3}+27\) needs to be rewritten in the form of \(a^{3}+b^{3}\). This can be done by taking a cube root of individual terms. The cube root of \(64x^{3}\) is \(4x\) and the cube root of \(27\) is \(3\). Hence the expression can be written as \((4x)^3+(3)^3\). After identifying \(a\) and \(b\), proceed to the next step.
02

Apply the Formula for the Sum of Cubes

Now, apply the formula for the sum of cubes, which is \(a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\). Here, \(a=4x\) and \(b=3\). By substituting these values in the formula, we get: \((4x+3)((4x)^{2}-(4x)(3)+(3)^{2})\)
03

Simplify the Result

Next, simplify the second part of the equation. The square of \(4x\) is \(16x^{2}\), the product of \(4x\) and \(3\) is \(12x\), and the square of \(3\) is \(9\). Therefore the expression simplifies to: \((4x+3)(16x^{2}-12x+9)\)

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