Chapter 4.1, Problem 32E

To determine

To write:a polynomial equation of least degree for $-2,3$ and find the degree as even or odd. Also find the times graph of the related function cross the x-axis.

Answer to Problem 32E

  $x^2-x-6=0$

The degree of this equation is 2, even degree

Since both the roots are real, the graph will cross the x-axis twice

Explanation of Solution

Given information: $-2,3$

Calculation:

If $x=-2,$ then

  $x+2\text{ is }$ a factor of the polynomial.

Likewise, if $x=3$ then $x-3$ is a factor of the polynomial.

Therefore, the linear factors for the polynomial are

  $\begin{array}{l}\left(x+2\right)\\ \left(x-3\right)\end{array}$

Now find the product of these factors.

  $\left(x+2\right)\left(x-3\right)=0$

Multiply the factors

  $x^2-x-6=0$

A polynomial equation with roots $-2,3$ is

  $x^2-x-6=0$

The degree of this equation is 2. Thus the equation has an even degree since 2 is an even number.

Since both the roots are real, the graph will cross the x-axis twice.