Chapter 5, Problem 12CT

Solution

To calculate : The roots of the given equation.

The roots of the given equation are $1,1,1,\text{ and }\frac{2}{3}$ .

Given information :

The equation is: $3x^4-11x^3+15x^2-9x+2=0$ .

Concept used:

Rational Root Theorem: For a given polynomial with integer coefficients, the rational roots will be in form of $\frac{p}{q}$ , where $p$ is the integer factor of the constant term and $q$ is the integer factor of the leading coefficient.

Calculation :

The degree of the equation $3x^4-11x^3+15x^2-9x+2=0$ is $4$ . So, it will have four roots.

Use synthetic division method to find the roots.

  $\begin{array}{l}\left\{\begin{array}{l}\overline{)1}3-1115-92\\ 3-87-2\\ \overline{3-87-2\overline{)0}}\end{array}\right\}\\ \left\{\begin{array}{l}\overline{)1}3-87-2\\ 3-52\\ \overline{3-52\overline{)0}}\end{array}\right\}\\ \left\{\begin{array}{l}\overline{)1}3-52\\ 3-2\\ \overline{3-2\overline{)0}}\end{array}\right\}\end{array}$

So,

  $3x^4-11x^3+15x^2-9x+2={\left(x-1\right)}^3\left(3x-2\right)$

As the remainder is equal to zero, so $1,1,\text{ and }1$ are the three roots of the given equation.

Find the fourth root.

  $\begin{array}{c}\left(3x-2\right)=0\\ x=\frac{2}{3}\end{array}$

Therefore, the roots of the given equation are: $1,1,1,\text{ and }\frac{2}{3}$ .