Expant uning the fornomial theor em \[ (3 x-1)^{5} \]

To expand \( (3x - 1)^5 \), we can use the Binomial Theorem, which states that \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, let \( a = 3x \), \( b = -1 \), and \( n = 5 \). The coefficients for each term will be given by the binomial coefficients \( \binom{5}{k} \). Expanding gives us: \[ (3x - 1)^5 = \sum_{k=0}^{5} \binom{5}{k} (3x)^{5-k} (-1)^k \] Calculating each term, we get: 1. For \( k = 0 \): \( \binom{5}{0}(3x)^5(-1)^0 = 243x^5 \) 2. For \( k = 1 \): \( \binom{5}{1}(3x)^4(-1)^1 = -405x^4 \) 3. For \( k = 2 \): \( \binom{5}{2}(3x)^3(-1)^2 = 270x^3 \) 4. For \( k = 3 \): \( \binom{5}{3}(3x)^2(-1)^3 = -90x^2 \) 5. For \( k = 4 \): \( \binom{5}{4}(3x)^1(-1)^4 = 15x \) 6. For \( k = 5 \): \( \binom{5}{5}(3x)^0(-1)^5 = -1 \) Putting it all together, we have: \[ (3x - 1)^5 = 243x^5 - 405x^4 + 270x^3 - 90x^2 + 15x - 1 \] Now, let's break it down a little more! Imagine you’re a mathematician in 19th century France when the Binomial Theorem was polished by Pascal. It was a breakthrough paving the way for modern algebra! The beauty of binomial expansions lay in their simplicity, growing complex like a blooming flower in a garden of mathematical wonders. In real-life applications, this theorem is invaluable! From computing probabilities to modeling phenomena in physics and finance, the ability to expand expressions allows for deeper analysis. It's like having a trusty toolbox—open it up, and you’ll find the instruments to solve challenges you're facing!