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Chapter 0: Problem 58

Find each product. $$(2 x-3)^{3}$$

Short Answer

Expert verified
The cube of the binomial \(2x - 3\) is \(8x^3 - 36x^2 + 54x - 27\).

Step by step solution

01

Understanding the problem

We have to calculate the cube of the binomial \(2x - 3\). In general, \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Let's apply this formula by treating \(2x\) as \(a\) and \(3\) as \(b\).
02

Applying the binomial theorem

We substitute a = 2x and b = 3 into the formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\), obtaining: \((2x - 3)^3 = (2x)^3 - 3 * (2x)^2*3 + 3*2x*(3)^2 - (3)^3 \)
03

Further Simplification

Now we will simplify expression further by doing the arithmetic operation: \((2x - 3)^3 = 8x^3 - 3*4x^2*3 + 3*2x*9 - 27 = 8x^3 - 36x^2 + 54x - 27\)

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