Chapter 9.2, Problem 35E

Solution

To identify: The coefficients in the polynomial expansion of ${\left(x-y\right)}^{50}$ alternate Is true or false.

The coefficient in the polynomial expansion ${\left(x-y\right)}^{50}$ alternate in sign is true.

Given information:

The given expression is ${\left(x-y\right)}^{50}$ .

Explanation:

Consider the given expression

Use the Binomial Theorem to expand ${\left(a+b\right)}^n={\displaystyle \sum _{r=0}^{n}c{}_r{a}^{n-r}{b}^r}$

  $\begin{array}{c}{\left(x+y\right)}^n={\displaystyle \sum _{r=0}^{50}c{}_r}{x}^{n-r}{\left(-y\right)}^r\\ {\left(x+y\right)}^n={\displaystyle \sum _{r=0}^{50}c{}_r}{x}^{n-r}{\left(-1\right)}^r{\left(y\right)}^r\end{array}$

When $r=0$ , then ${\left(-1\right)}^r={\left(-1\right)}^0=1$ , positive.

When $r=1$ , then ${\left(-1\right)}^r={\left(-1\right)}^1=-1$ , negative.

When $r=2$ , then ${\left(-1\right)}^r={\left(-1\right)}^2=1$ , positive.

Then third term is positive.

By continuing this, alternate sign in the expansion of ${\left(x-y\right)}^{50}$ .

Hence, the coefficient in the polynomial expansion ${\left(x-y\right)}^{50}$ alternate in sign is true.