Chapter 5.7, Problem 3E

To determine

To graph: the given function, identify real zeros, maximum and minimum points, number and types of zeroes, y-intercepts, and symmetry.

Answer to Problem 3E

Zeros of given polynomial $k(x)$ are 0, 1, -2

Maximum at points= (-1.443, 2.833), (0.693, 0.397)

Minimum at point= (0, 0)

Number of zeroes= 3

All three zeros are real and there is no symmetry.

Y-intercepts= 0.

Explanation of Solution

Given information: a polynomial is given as follows-

  $k(x)=-x^4-x^3+2x^2$

Graph:

  

To find the zeros polynomial equate $k(x)=0$

  $\begin{array}{l}\Rightarrow -x^4-x^3+2x^2=0\\ \Rightarrow -x^2(x^2+x-2)=0\\ \Rightarrow -x^2\{x^2+2x-x-2\}=0\\ \Rightarrow -x^2\{x(x+2)-(x+2)\}=0\\ \Rightarrow -x^2(x+2)(x-1)=0\\ \Rightarrow -x^2=0\text{ or }x+2=0\text{ or }x-1=0\\ \Rightarrow x=0,-2,1\end{array}$

Clearly from the graph $k(x)$ is maximum at points= (-1.443, 2.833), (0.693, 0.397)

Clearly from the graph $k(x)$ is minimum at point= (0, 0).

All the roots are real and there is symmetry about any line.

Since it crosses the y-axis at (0, 0), so the y-intercepts=0