Verified Answer :
- Use the Binomial Theorem with \(n=5\) to expand the expression \((3s+2)^5\).
- Retrieve the coefficients from the 5th row of Pascal's Triangle: \(1, 5, 10, 10, 5, 1\).
- Apply the formula \((3s+2)^5 = \sum_{k=0}^{5} \binom{5}{k} (3s)^{5-k} (2)^k\) to compute each term.
- Result in: \(\boxed{243s^5 + 810s^4 + 1080s^3 + 720s^2 + 240s + 32}\).
Explanation
- Understanding the Binomial Expansion and Pascal's Triangle
To expand \((3s+2)^5\), we utilize the Binomial Theorem, which states that for any non-negative integer \(n\), \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). In this specific case, \(n=5\), \(x=3s\), and \(y=2\). The coefficients \(\binom{n}{k}\) can be obtained from the 5th row of Pascal's Triangle, which are 1, 5, 10, 10, 5, and 1.
- Setting up the Binomial Expansion
We set up the expansion as follows:
\((3s+2)^5 = \binom{5}{0}(3s)^5(2)^0 + \binom{5}{1}(3s)^4(2)^1 + \binom{5}{2}(3s)^3(2)^2 + \binom{5}{3}(3s)^2(2)^3 + \binom{5}{4}(3s)^1(2)^4 + \binom{5}{5}(3s)^0(2)^5\).
Substituting the coefficients from Pascal's Triangle (1, 5, 10, 10, 5, 1):
\((3s+2)^5 = 1 \cdot (3s)^5 \cdot 1 + 5 \cdot (3s)^4 \cdot 2 + 10 \cdot (3s)^3 \cdot 4 + 10 \cdot (3s)^2 \cdot 8 + 5 \cdot (3s)^1 \cdot 16 + 1 \cdot 1 \cdot 32\).
- Calculating and Combining the Terms
Now we compute each individual term:
- First term: \(1 \cdot 243s^5 \cdot 1 = 243s^5\)
- Second term: \(5 \cdot 81s^4 \cdot 2 = 810s^4\)
- Third term: \(10 \cdot 27s^3 \cdot 4 = 1080s^3\)
- Fourth term: \(720s^2\)
- Fifth term: \(5 \cdot 3s \cdot 16 = 240s\)
- Sixth term: \(1 \cdot 1 \cdot 32 = 32\)
Combining these results gives the full expanded polynomial.
- Conclusion
The final expanded polynomial for \((3s+2)^5\) is \(243s^5 + 810s^4 + 1080s^3 + 720s^2 + 240s + 32\). Thus, \((3s+2)^5 = \boxed{243s^5 + 810s^4 + 1080s^3 + 720s^2 + 240s + 32}\).
Examples
Binomial expansion is essential in various fields such as finance, where it helps calculate the future value of investments over multiple periods, and physics, where it is used to approximate complex functions in problems involving series. For instance, if you're predicting compound interest growth for an investment over five stages with a constant rate, binomial expansion provides a quick, systematic way to determine the total accumulation. Understanding how to expand binomials efficiently allows for faster problem-solving in data modeling and statistical analysis.