Chapter 8.7, Problem 7WE

To determine

To find the rational roots of the equation.

Answer to Problem 7WE

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

Explanation of Solution

Given information:

  $2x^3+7x^2+6x-5=0$

Calculation:

The possible rational roots can be obtained as

  $2x^3+7x^2+6x-5=0$

First find h and k

  $\begin{array}{l}h=1\text{ or 5 }\left[\begin{array}{l}h=\text{factor of the constant term of the polynomial}\\ \text{since, 5 is divisible by 1 and 5}\end{array}\right]\\ k=1\text{ or }2\left[\begin{array}{l}\text{k= factor of the cofficient of the highest-degree term}\\ \text{since, 2 is divisible by 1 and 2}\end{array}\right]\end{array}$

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

Therefore,

  $\pm \frac{1}{1}=\pm 1\text{ or }\pm \frac{5}{1}=\pm 5\text{ or }\pm \frac{1}{2}=\pm \frac{1}{2}\text{ or }\pm \frac{5}{2}=\pm \frac{5}{2}$

  $\begin{array}{l}\overline{)1}\text{ 2 7 6 }-5\\ \underset{\_}{\text{ 2 9 15}}\\ \text{ 2 9 15 }⋮\text{ 10}\leftarrow \text{ 1 is not a root}\\ \overline{)-1}\text{ 2 7 6 }-5\\ \underset{\_}{-\text{2 }-5-1}\\ \text{ 2 5 1 }⋮-6\leftarrow -\text{1 is not a root}\\ \overline{)5}\text{ 2 7 6 }-5\\ \underset{\_}{\text{ 10 85 455}}\\ \text{ 2 17 91 }⋮\text{ 450}\leftarrow \text{ 5 is not a root}\\ \overline{)-5}\text{ 2 7 6 }-5\\ \underset{\_}{-10\text{ 15 }-\text{105}}\\ \text{ 2 }-3\text{ 21 }⋮-1\text{10}\leftarrow -5\text{ is not a root}\\ \overline{)\frac{1}{2}}\text{ 2 7 6 }-5\\ \underset{\_}{\text{ 1 4 5}}\\ \text{ 2 8 10 }⋮\text{ 0}\leftarrow \frac{1}{2}\text{ is a root}\end{array}$

The depressed equation is $2x^2+8x+10=0$

Apply Sridhar Acharya formula to solve the above equation

  $\begin{array}{l}2x^2+8x+10=0\\ x=\frac{-\left(8\right)\pm \sqrt{{\left(8\right)}^2-4\left(2\right)\left(10\right)}}{2\left(2\right)}\\ x=\frac{-8\pm \sqrt{64-80}}{4}\\ x=\frac{-8\pm \sqrt{-16}}{4}\\ x=\frac{-8\pm 4i}{4}\left[\text{since, }{i}^2=-1\right]\\ x=-2\pm i\left[\text{divide numerator and denominator by 4}\right]\end{array}$

Hence,

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