Chapter 4, Problem 13CR

To determine

To calculate: The expansion of the binomial using Pascal’s Triangle.

Answer to Problem 13CR

The required expansion is $243s^5+810s^4+1080s^3+720s^2+240s+32$ .

Explanation of Solution

Given information:

The expression: ${\left(3s+2\right)}^5$

Calculation:

The given expression is ${\left(3s+2\right)}^5$ .

Use Pascal’s Triangle on the above expression.

  $\begin{array}{l}1\\ 11\\ 121\\ 1331\\ 14641\\ \boxed{15101051}\end{array}$

From the above Pascal’s Triangle, write the expansion of the given expression as:

  $\begin{array}{c}{\left(3m+2\right)}^5={\left(3s\right)}^5+5{\left(3s\right)}^4\left(2\right)+10{\left(3s\right)}^3{\left(2\right)}^2+10{\left(3s\right)}^2{\left(2\right)}^3+5\left(3s\right){\left(2\right)}^4+2^5\\ {\left(3m+2\right)}^5=243s^5+10\left(81s^4\right)+10\left(27s^3\right)\left(4\right)+15s+32\\ {\left(3m+2\right)}^5=243s^5+810s^4+1080s^3+720s^2+240s+32\end{array}$

Hence, the required expansion is $243s^5+810s^4+1080s^3+720s^2+240s+32$ .