Chapter 9.5, Problem 32E

To determine

To expand: The expression ${\left(2t-s\right)}^5$ by using Pascal’s triangle to determine the coefficients.

Answer to Problem 32E

The expansion of the expression ${\left(2t-s\right)}^5$ is $32t^5-80t^{4}s+80t^3{s}^2-40t^2{s}^3+10t{s}^4-s^5$ .

Explanation of Solution

Given information: The expression is ${\left(2t-s\right)}^5$ .

Formula used:Binomial Theorem: ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\cdots +nx{y}^{n-1}+y^n$ .

Calculation:

Here, $n=5$ .

First write out the Pascal’s triangle such that the row that begins with 1, 5.

  $\begin{array}{ccccccccccc}& & & & & 1& & & & & \\ & & & & 1& & 1& & & & \\ & & & 1& & 2& & 1& & & \\ & & 1& & 3& & 3& & 1& & \\ & 1& & 4& & 6& & 4& & 1& \\ 1& & 5& & 10& & 10& & 5& & 1\end{array}$

From the Pascal’s triangle, it is observed that the coefficients of the expression are 1, 5, 10, 10, 5 and 1.

Expand the expression ${\left(2t-s\right)}^5$ by using formula in Binomial theorem as follows.

  $\begin{array}{c}{\left(2t-s\right)}^5=\left(1\right){\left(2t\right)}^5+5{\left(2t\right)}^4\left(-s\right)+10{\left(2t\right)}^3{\left(-s\right)}^2+10{\left(2t\right)}^2{\left(-s\right)}^3+5{\left(2t\right)}^1{\left(-s\right)}^4+1{\left(-s\right)}^5\\ =32t^5-80t^{4}s+80t^3{s}^2-40t^2{s}^3+10t{s}^4-s^5\end{array}$

Thus, the expansion of the expression ${\left(2t-s\right)}^5$ is $32t^5-80t^{4}s+80t^3{s}^2-40t^2{s}^3+10t{s}^4-s^5$ .