Chapter 6, Problem 4REAE

Solution

To write: the binomial expansion using the binomial theorem.

  ${\left(r^3-4s\right)}^5=r^{15}-20r^{12}s+160r^9{s}^2-640r^6{s}^3+1280r^3{s}^4-1024s^5$

Given information:

The given expression is:

  ${\left(r^3-4s\right)}^5$

Formula Used:

Combinations of n Objects Taken r at a Time

The number of combinations of r objects taken from a group of n distinct objects is denoted by $C{}_r$ and is given by the formula:

  $C{}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$ .

  $\begin{array}{l}C{}_0=1\\ C{}_n=1\end{array}$

Theorem Used:

Binomial Theorem

For any positive integer n , the binomial expansion of ${\left(a+b\right)}^n$ is:

  ${\left(a+b\right)}^n=C{}_0{a}^n{b}^0+C{}_1{a}^{n-1}{b}^1+C{}_2{a}^{n-2}{b}^2+\cdots +C{}_n{a}^0{b}^n$

Here, each term in the expansion of ${\left(a+b\right)}^n$ has the form $C{}_r{a}^{n-r}{b}^r$ where r s an integer from 0 to n .

Explanation:

By using the binomial theorem, the given expression ${\left(r^3-4s\right)}^5$ is written in expanded form as:

  ${\left(r^3-4s\right)}^5={\left(r^3+\left(-4s\right)\right)}^5$

  $\begin{array}{c}=C{}_0{\left(r^3\right)}^5{\left(-4s\right)}^0+C{}_1{\left(r^3\right)}^4{\left(-4s\right)}^1+C{}_2{\left(r^3\right)}^3{\left(-4s\right)}^2+C{}_3{\left(r^3\right)}^2{\left(-4s\right)}^3\\ +C{}_4{\left(r^3\right)}^1{\left(-4s\right)}^4+C{}_5{\left(r^3\right)}^0{\left(-4s\right)}^5\end{array}$

  $\begin{array}{c}=\left(\frac{5!}{\left(5-0\right)!0!}\right)\left(r^{15}\right)\left(1\right)+\left(\frac{5!}{\left(5-1\right)!1!}\right)\left(r^{12}\right)\left(-4s\right)+\left(\frac{5!}{\left(5-2\right)!2!}\right)\left(r^9\right)\left(16s^2\right)\\ +\left(\frac{5!}{\left(5-3\right)!3!}\right)\left(r^6\right)\left(-64s^3\right)+\left(\frac{5!}{\left(5-4\right)!4!}\right)\left(r^3\right)\left(256s^4\right)+\left(\frac{5!}{\left(5-5\right)!5!}\right)\left(1\right)\left(-1024s^5\right)\end{array}$

  $\begin{array}{c}=1\left(r^{15}\right)\left(1\right)+\left(5\right)\left(r^{12}\right)\left(-4s\right)+\left(10\right)\left(r^9\right)\left(16s^2\right)+\left(10\right)\left(r^6\right)\left(-64s^3\right)\\ +\left(5\right)\left(r^3\right)\left(256s^4\right)+\left(1\right)\left(1\right)\left(-1024s^5\right)\\ =r^{15}-20r^{12}s+160r^9{s}^2-640r^6{s}^3+1280r^3{s}^4-1024s^5\end{array}$