Chapter 8.3, Problem 56E

a.

To determine

Write each polynomial as a rational expression by using rule of finite geometric series.

Answer to Problem 56E

  $\boxed{S_5=\frac{1-x^5}{1-x}}$

Explanation of Solution

Given information:

  $1+x+x^2+x^3+x^4$

Formula used:

Sum of n term in geometric series $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$

First term of the series $a_1=1$

Common ratio:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{x}{1}=x\\ r=x\end{array}$

The polynomial is the sum of the first 5 terms of a geometric sequence with the elements:

Here we can see there are 5 terms in polynomial, therefore $n=5,a=1,r=x$ ,

  $\begin{array}{l}{S}_n=\frac{a_1\left(1-r^n\right)}{1-r}\\ \\ S_5=\frac{1\left(1-x^5\right)}{1-x}\\ \\ \boxed{S_5=\frac{1-x^5}{1-x}}\end{array}$

b.

To determine

Write each polynomial as a rational expression by using rule of finite geometric series.

Answer to Problem 56E

  $\boxed{S_5=\frac{3x-96x^6}{1-2x^2}}$

Explanation of Solution

Given information:

  $3x+6x^3+12x^5+24x^7$

Formula used:

Sum of n term in geometric series $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$

First term of the series $a_1=3x$

Common ratio:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{6x^3}{3x}=2x^2\\ r=2x^2\end{array}$

The polynomial is the sum of the first 5 terms of a geometric sequence with the elements:

Here we can see there are 5 terms in polynomial, therefore $n=5,a=3x,r=2x^2$ ,

  $\begin{array}{l}{S}_n=\frac{a_1\left(1-r^n\right)}{1-r}\\ \\ S_5=\frac{3x\left(1-{\left(2x^2\right)}^5\right)}{1-2x^2}\\ \\ S_5=\frac{3x\left(1-32x^5\right)}{1-2x^2}\\ \\ \boxed{S_5=\frac{3x-96x^6}{1-2x^2}}\end{array}$