Chapter 6.2, Problem 6Q

Solution

To write: the binomial expansion using the binomial theorem.

  ${\left(2s-3\right)}^6=64s^6-576s^5+2160s^4-4320s^3+4860s^2-2916s+729$

Given information:

The given expression is:

  ${\left(2s-3\right)}^6$

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 ${\left(2s-3\right)}^6$ is written in expanded form as:

  ${\left(2s-3\right)}^6={\left(2s+\left(-3\right)\right)}^6$

  $\begin{array}{c}=C{}_0{\left(2s\right)}^6{\left(-3\right)}^0+C{}_1{\left(2s\right)}^5{\left(-3\right)}^1+C{}_2{\left(2s\right)}^4{\left(-3\right)}^2+C{}_3{\left(2s\right)}^3{\left(-3\right)}^3\\ +C{}_4{\left(2s\right)}^2{\left(-3\right)}^4+C{}_5{\left(2s\right)}^1{\left(-3\right)}^5+C{}_6{\left(2s\right)}^0{\left(-3\right)}^6\end{array}$

  $\begin{array}{c}=\left(\frac{6!}{\left(6-0\right)!0!}\right)\left(64s^6\right)\left(1\right)+\left(\frac{6!}{\left(6-1\right)!1!}\right)\left(32s^5\right)\left(-3\right)+\left(\frac{6!}{\left(6-2\right)!2!}\right)\left(16s^4\right)\left(9\right)\\ +\left(\frac{6!}{\left(6-3\right)!3!}\right)\left(8s^3\right)\left(-27\right)+\left(\frac{6!}{\left(6-4\right)!4!}\right)\left(4s^2\right)\left(81\right)+\left(\frac{6!}{\left(6-5\right)!5!}\right)\left(2s^1\right)\left(-243\right)\end{array}$

  $\begin{array}{c}+\left(\frac{6!}{\left(6-6\right)!6!}\right)\left(1\right)\left(729\right)\\ =1\left(64s^6\right)\left(1\right)+\left(6\right)\left(32s^5\right)\left(-3\right)+\left(15\right)\left(16s^4\right)\left(9\right)+\left(20\right)\left(8s^3\right)\left(-27\right)\\ +\left(15\right)\left(4s^2\right)\left(81\right)+\left(6\right)\left(2s^1\right)\left(-243\right)+\left(1\right)\left(1\right)\left(729\right)\\ =64s^6-576s^5+2160s^4-4320s^3+4860s^2-2916s+729\end{array}$