Chapter 6, Problem 2REAE

Solution

To write: the binomial expansion using the binomial theorem.

  ${\left(2a+b^2\right)}^4=16a^4+32a^3{b}^2+24a^2{b}^4+8a{b}^6+b^8$

Given information:

The given expression is:

  ${\left(2a+b^2\right)}^4$

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

  $\begin{array}{c}{\left(2a+b^2\right)}^4=C{}_0{\left(2a\right)}^4{\left(b^2\right)}^0+C{}_1{\left(2a\right)}^3{\left(b^2\right)}^1+C{}_2{\left(2a\right)}^2{\left(b^2\right)}^2+C{}_3{\left(2a\right)}^1{\left(b^2\right)}^3+C{}_4{\left(2a\right)}^0{\left(b^2\right)}^4\\ =\left(\frac{4!}{\left(4-0\right)!0!}\right)\left(16a^4\right)\left(1\right)+\left(\frac{4!}{\left(4-1\right)!1!}\right)\left(8a^3\right)\left(b^2\right)+\left(\frac{4!}{\left(4-2\right)!2!}\right)\left(4a^2\right)\left(b^4\right)\\ +\left(\frac{4!}{\left(4-3\right)!3!}\right)\left(2a^1\right)\left(b^6\right)+\left(\frac{4!}{\left(4-4\right)!4!}\right)\left(1\right)\left(b^8\right)\\ =1\left(16a^4\right)\left(1\right)+\left(4\right)\left(8a^3\right)\left(b^2\right)+\left(6\right)\left(4a^2\right)\left(b^4\right)+\left(4\right)\left(2a\right)\left(b^6\right)+\left(1\right)\left(1\right)\left(b^8\right)\\ =16a^4+32a^3{b}^2+24a^2{b}^4+8a{b}^6+b^8\end{array}$