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Chapter 11: Problem 14

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (4 x-1)^{3} $$

Short Answer

Expert verified
The simplified form of the binomial expansion \((4x - 1)^3\) is \(64x^3 - 48x^2 + 12x - 1\).

Step by step solution

01

- Identify the Parameters

We identify the parameters according to the Binomial Theorem. In the exercise \((4x - 1)^3\), \(a = 4x\), \(b = -1\), and \(n = 3\).
02

- Apply the Binomial Theorem Formula

We apply the Binomial Theorem formula: \[ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^{k}\]. By substituting the identified parameters into the formula, we obtain: \[ (4x - 1)^3 = {3 \choose 0}(4x)^{3-0}(-1)^{0} + {3 \choose 1}(4x)^{3-1}(-1)^{1} + {3 \choose 2}(4x)^{3-2}(-1)^{2} + {3 \choose 3}(4x)^{3-3}(-1)^{3} \]
03

- Simplify the Expansion

Evaluate the factorial components, simplify powers, and perform multiplications. The expanded form of the binomial becomes \[ 64x^3 - 48x^2 + 12x - 1 \]

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