Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Identify the Parameters

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

Step 2 - 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: \[ (4x - 1)^3 = {3 \choose 0}(4x)^{3-0}(-1)^{0} + {3 \choose 1}(4x)^{3-1}(-1)^{1} + {3 \choose 2}(4x)^{3-2}(-1)^{2} + {3 \choose 3}(4x)^{3-3}(-1)^{3} \]

Step 3 - Simplify the Expansion

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