Chapter 4.3, Problem 47E

(a)

To determine

To find: whether the given numbers is the root of the equation.

Answer to Problem 47E

The value $2$ is not the root of the equation.

Explanation of Solution

Given:

  $x^4-4x^3-x^2+4x=0$ .

Concept used:

The root of a polynomial are those values of the variable that cause the polynomial to evaluate to zero.

Calculation:

Consider the equation:

  $x^4-4x^3-x^2+4x=0$ .

Let $f\left(x\right)=x^4-4x^3-x^2+4x$ .

To determine whether $2$ is a root of the equation, first evaluate $f\left(2\right)$ .

  $f\left(x\right)=x^4-4x^3-x^2+4x$ .

Write the original polynomial.

  $\begin{array}{l}f\left(2\right)={\left(2\right)}^4-4{\left(2\right)}^3-{\left(2\right)}^2+4\left(2\right).\\ \Rightarrow 16-32-4+8.\\ \Rightarrow -12.\end{array}$

Hence, $2$ is not the root of the equation.

(b)

To determine

To find: whether the given numbers is the root of the equation.

Answer to Problem 47E

  $0$ is the root of the equation.

Explanation of Solution

Given:

  $x^4-4x^3-x^2+4x=0$ .

Concept used:

The root of a polynomial are those values of the variable that cause the polynomial to evaluate to zero.

Calculation:

Consider the equation:

  $x^4-4x^3-x^2+4x=0$ .

Let $f\left(x\right)=x^4-4x^3-x^2+4x$ .

To determine whether $0$ is a root of the equation, first evaluate $f\left(0\right)$ .

  $f\left(x\right)=x^4-4x^3-x^2+4x$ .

Write the original polynomial and substitute $x=0$ :

  $\begin{array}{l}f\left(0\right)={\left(0\right)}^4-4{\left(0\right)}^3-{\left(0\right)}^2+4\left(0\right)\\ \Rightarrow 0-0-0+0\\ \Rightarrow 0.\\ f\left(0\right)=0.\end{array}$

Hence, $0$ is the root of the equation.

(c)

To determine

To find: whether the given numbers is the root of the equation.

Answer to Problem 47E

Range is $\{g(x)|g(x)>0\}$ .

Explanation of Solution

Given:

  $x^4-4x^3-x^2+4x=0$ .

Concept used:

The root of a polynomial are those values of the variable that cause the polynomial to evaluate to zero.

Calculation:

Consider the equation:

  $x^4-4x^3-x^2+4x=0$ .

Let $f\left(x\right)=x^4-4x^3-x^2+4x$ .

To determine whether $-2$ is a root of the equation, first evaluate $f\left(-2\right)$ .

  $f\left(x\right)=x^4-4x^3-x^2+4x$ .

Write the original polynomial and substitute $x=-2$ :

  $\begin{array}{l}f\left(-2\right)={\left(-2\right)}^4-4{\left(-2\right)}^3-{\left(-2\right)}^2+4\left(-2\right)\\ \Rightarrow 16+32-4-8\\ \Rightarrow 36.\\ f\left(-2\right)=36.\end{array}$

Hence, $-2$ is the root of the equation.

(d)

To determine

To find: whether the given numbers is the root of the equation.

Answer to Problem 47E

  $4$ is the root of the equation.

Explanation of Solution

Given:

  $x^4-4x^3-x^2+4x=0$ .

Concept used:

The root of a polynomial are those values of the variable that cause the polynomial to evaluate to zero.

Calculation:

Consider the equation:

  $x^4-4x^3-x^2+4x=0$ .

Let $f\left(x\right)=x^4-4x^3-x^2+4x$ .

To determine whether $4$ is a root of the equation, first evaluate $f\left(4\right)$ .

  $f\left(x\right)=x^4-4x^3-x^2+4x$ .

Write the original polynomial and substitute $x=4$ :

  $\begin{array}{l}f\left(4\right)={\left(4\right)}^4-4{\left(4\right)}^3-{\left(4\right)}^2+4\left(4\right)\\ \Rightarrow 256-256-16+16\\ \Rightarrow 0.\\ f\left(4\right)=0.\end{array}$

Hence, $4$ is the root of the equation.