Chapter 8.7, Problem 10WE

To determine

To find the rational roots of the equation.

Answer to Problem 10WE

The rational roots are $-\frac{1}{2},-\frac{1}{2},\pm 2i$

Explanation of Solution

Given information:

  $4x^4+4x^3+17x^2+16x+4=0$

Calculation:

The possible rational roots can be obtained as

  $4x^4+4x^3+17x^2+16x+4=0$

First find h and k

  $\begin{array}{l}h=1,2\text{ or 4 }\left[\begin{array}{l}h=\text{factor of the constant term of the polynomial}\\ \text{since, 4 is divisible by 1,2 or 4}\end{array}\right]\\ k=1,2\text{ or 4 }\left[\begin{array}{l}\text{k= factor of the cofficient of the highest-degree term}\\ \text{since, 4 is divisible by 1,2 or 4}\end{array}\right]\end{array}$

  $$ All possible rational roots are in the form of $\pm \frac{h}{k}$

Therefore,

  $\begin{array}{l}\pm \frac{1}{1}=\pm 1\text{ or }\pm \frac{2}{1}=\pm 2\text{ or }\pm \frac{4}{1}=\pm 4\\ \pm \frac{1}{2}=\pm \frac{1}{2}\text{ or }\pm \frac{2}{2}=\pm 1\text{ or }\pm \frac{4}{2}=\pm 2\\ \pm \frac{1}{4}=\pm \frac{1}{4}\text{ or }\pm \frac{2}{4}=\pm \frac{1}{2}\text{ or }\pm \frac{4}{4}=\pm 1\end{array}$

  $\begin{array}{l}\overline{)-1}\text{ 4 4 17 16 4}\\ \underset{\_}{-4\text{ 0 }-17\text{ 1}}\\ \text{ 4 0 17 }-1⋮\text{ 5}\leftarrow -1\text{ is not a root}\\ \overline{)1}\text{ 4 4 17 16 4}\\ \underset{\_}{4\text{ 8 25 41}}\\ \text{ 4 8 25 41 }⋮\text{ 45}\leftarrow 1\text{ is not a root}\\ \overline{)-2}\text{ 4 4 17 16 4}\\ \underset{\_}{-8\text{ 8 }-50\text{ 68}}\\ \text{ 4 }-4\text{ 25 }-34⋮\text{ 72}\leftarrow -2\text{ is not a root}\\ \overline{)-4}\text{ 4 4 17 16 4}\\ \underset{\_}{-16\text{ 48 }-260\text{ 976}}\\ \text{ 4 }-12\text{ 65 }-244⋮\text{ 980}\leftarrow -4\text{ is not a root}\\ \overline{)-\frac{1}{2}}\text{ 4 4 17 16 4}\\ \underset{\_}{-2-1-8-4}\\ \text{ 4 2 16 8 }⋮\text{ 0}\leftarrow -\frac{1}{2}\text{ is a root}\end{array}$

The depressed equation is $4x^3+2x^2+16x+8=0$

  $\begin{array}{l}\overline{)2}\text{ 4 2 16 8}\\ \underset{\_}{8\text{ 20 72}}\\ \text{ 4 10 36 }⋮\text{ 80}\leftarrow 2\text{ is not a root}\\ \overline{)4}\text{ 4 2 16 8}\\ \underset{\_}{16\text{ 72 352 }}\\ \text{ 4 }18\text{ 88 }⋮\text{ 360}\leftarrow 4\text{ is not a root}\\ \overline{)-\frac{1}{2}}\text{ 4 2 16 8}\\ \underset{\_}{-2\text{ 0 }-8}\\ \text{ 4 0 16 }⋮\text{ 0}\leftarrow -\frac{1}{2}\text{ is a root}\end{array}$

Again the depressed equation is $4x^2+16=0$

  $\begin{array}{l}4x^2+16=0\\ 4x^2=-16\\ x^2=-4\left[\text{divide both sides by 4}\right]\\ x=\pm 2i\left[\text{write }{i}^2=-1\right]\end{array}$

Hence,

The rational roots are $-\frac{1}{2},-\frac{1}{2},\pm 2i$