Chapter 6.1, Problem 29E

Solution

To write: the binomial expansion using the binomial theorem.

  ${\left(2s^4+5\right)}^5=32s^{20}+400s^{16}+2000s^{12}+5000s^8+6250s^4+3125$

Given information:

The given expression is:

  ${\left(2s^4+5\right)}^5$

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 ${\left(2s^4+5\right)}^5$ is written in expanded form as:

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

  $\begin{array}{c}=\left(\frac{5!}{\left(5-0\right)!0!}\right)\left(32s^{20}\right)\left(5^0\right)+\left(\frac{5!}{\left(5-1\right)!1!}\right)\left(16s^{16}\right)\left(5\right)+\left(\frac{5!}{\left(5-2\right)!2!}\right)\left(8s^{12}\right)\left(5^2\right)\\ +\left(\frac{5!}{\left(5-3\right)!3!}\right)\left(4s^2\right)\left(5^3\right)+\left(\frac{5!}{\left(5-4\right)!4!}\right)\left(2s^1\right)\left(5^4\right)+\left(\frac{5!}{\left(5-5\right)!5!}\right)\left(2^0\cdot s^0\right)\left(5^5\right)\end{array}$

  $\begin{array}{c}=1\left(32s^{20}\right)\left(1\right)+\left(5\right)\left(16s^{16}\right)\left(5\right)+\left(10\right)\left(8s^{12}\right)\left(25\right)+\left(10\right)\left(4s^2\right)\left(125\right)\\ +\left(5\right)\left(2s\right)\left(625\right)+\left(1\right)\left(1\right)\left(3125\right)\\ =32s^{20}+400s^{16}+2000s^{12}+5000s^8+6250s^4+3125\end{array}$