Chapter 9, Problem 20RE

Solution

To identify: The expansion of the binomial ${\left(4a-3b\right)}^7$ .

The expanded form of the binomial is

  $\begin{array}{l}{\left(4a-3b\right)}^7=16384a^7+86016a^{6}b+194536a^5{b}^2-241920a^4{b}^3+181440a^3{b}^4+-81648a^2{b}^5\\ +20412a{b}^6-2187b^7\end{array}$

Given information:

The given expression is ${\left(4a-3b\right)}^7$ .

Explanation:

Consider the given expression

The binomial Theorem state that for any positive integer

  $n,$ ${\left(a+b\right)}^n=\left(\begin{array}{l}n\\ 0\end{array}\right)a^n+\left(\begin{array}{l}n\\ 1\end{array}\right)a^{n-1}b+\mathrm{......}+\left(\begin{array}{l}n\\ r\end{array}\right)a^{n-r}{b}^r+\mathrm{.....}+\left(\begin{array}{l}n\\ n\end{array}\right)b^n$

Where $a=4a,n=7\text{and}b=-3b$

  $\begin{array}{c}{\left(4a-3b\right)}^7=\left(\begin{array}{l}7\\ 0\end{array}\right){\left(4a\right)}^7+\left(\begin{array}{l}7\\ 1\end{array}\right){\left(4a\right)}^6\left(-3b\right)+\left(\begin{array}{l}7\\ 2\end{array}\right){\left(4a\right)}^5{\left(-3b\right)}^2+\left(\begin{array}{l}7\\ 3\end{array}\right){\left(4a\right)}^4{\left(-3b\right)}^3+\left(\begin{array}{l}7\\ 4\end{array}\right){\left(4a\right)}^3{\left(-3b\right)}^4\\ +\left(\begin{array}{l}7\\ 5\end{array}\right){\left(4a\right)}^2{\left(-3b\right)}^5+\left(\begin{array}{l}7\\ 6\end{array}\right){\left(4a\right)}^1{\left(-3b\right)}^6+\left(\begin{array}{l}7\\ 7\end{array}\right){\left(-3b\right)}^7\\ {\left(4a-3b\right)}^7=1{\left(4a\right)}^7+7{\left(4a\right)}^6\left(-3b\right)+21{\left(4a\right)}^5{\left(-3b\right)}^2+35{\left(4a\right)}^4{\left(-3b\right)}^3+35{\left(4a\right)}^3{\left(y\right)}^4\\ +21{\left(4a\right)}^2{\left(-3b\right)}^5+7\left(4a\right){\left(-3b\right)}^{6}1{\left(-3b\right)}^7\\ {\left(4a-3b\right)}^7=16384a^7+86016a^{6}b+194536a^5{b}^2-241920a^4{b}^3+181440a^3{b}^4-81648a^2{b}^5\\ +20412a{b}^6-2187b^7\end{array}$ .

So, the expression.

  $\begin{array}{c}{\left(4a-3b\right)}^7=16384a^7+86016a^{6}b+194536a^5{b}^2-241920a^4{b}^3+181440a^3{b}^4+-81648a^2{b}^5\\ +20412a{b}^6-2187b^7\end{array}$