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Chapter 11: Problem 13

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

Short Answer

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

Step by step solution

01

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 \(n = 3\).
02

Apply the Binomial Theorem

Applying the Binomial Theorem now gives us: \((5x-1)^3 = \binom{3}{0}(5x)^3(-1)^0 + \binom{3}{1}(5x)^2(-1)^1 + \binom{3}{2}(5x)^1(-1)^2 + \binom{3}{3}(5x)^0(-1)^3\)
03

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: \(125x^3 - 3*25x^2 + 3*5x -1 \)
04

Simplify the Expression

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

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