Chapter 11.7, Problem 14WE

To determine

To Find: The last three terms of the form ${(p-q)}^{22}$ .

Answer to Problem 14WE

The required last three terms of the binomial form ${(p-q)}^{22}$ is $231p^2{q}^2-22p{q}^{21}+q^{22}$ .

Explanation of Solution

Given Information:

The first three terms are $p^{22}-22p^{21}q+231p^{20}{q}^2$ .

When the first and last terms have the same coefficients, this means symmetry of the coefficients, the second to last and the second terms also have the same coefficients, and the third to last and third have the same coefficients.

It is also to be noted that the powers of the binomial components are also switched symmetrically. The alternating minus sign is not at all treated as a part of the coefficient.

So, the alternating minus sign will come about because of changing power on $(-q)$ .

Therefore, the last three terms are,

  $\begin{array}{l}$ 231p^2{(-q)}^{20}+22p{(-q)}^{21}+{(-q)}^{22}$231p^2{q}^2-22p{q}^{21}+q^{22}\end{array}$

Hence, the last three terms of the binomial form ${(p-q)}^{22}$ is $231p^2{q}^2-22p{q}^{21}+q^{22}$ .