Problem 13 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (5 x-1)^{3} $$

Step-by-Step Solution

Answer

The expanded form of \((5x-1)^3\) is \(125x^3 - 75x^2 + 15x -1\).

Step 1: Identify the Components of the Binomial

Here, our binomial is \((5x-1)\) and it's raised to the power of 3. So our \(a = 5x\), \(b = -1\), and \(n = 3\).

Step 2: Apply the Binomial Theorem

Applying the Binomial Theorem now gives us: \((5x-1)^3 = \binom{3}{0}(5x)^3(-1)^0 + \binom{3}{1}(5x)^2(-1)^1 + \binom{3}{2}(5x)^1(-1)^2 + \binom{3}{3}(5x)^0(-1)^3\)

Step 3: Calculate the Binomial Coefficients and Powers

The Binomial coefficient, \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\). Using this, we calculate the coefficients and powers: \(125x^3 - 3*25x^2 + 3*5x -1 \)

Step 4: Simplify the Expression

Finally, we simplify the expression: \(125x^3 - 75x^2 + 15x -1\)