Chapter 4.1, Problem 9CFU

To determine

To write: a polynomial equation of least degree for $-5,7$ .

Answer to Problem 9CFU

A polynomial equation with roots $-5,7$ is $x^2-2x-35=0$

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

Explanation of Solution

Given information: $-5,7$

Calculation:

If $x=-5,\text{ then }x+5$ is a factor of the polynomial

Likewise, if $x=7,\text{ then }x-7$ is a factor of the polynomial

Therefore, the linear factors for the polynomial are $x+5\text{ and }x-7$ .

Now find the products of these factors.

  $\left(x+5\right)\left(x-7\right)=0$

Multiply these factors

  $x^2-2x-35=0$

Your solution

A polynomial equation with roots $-5,7$ is

  $x^2-2x-35=0$

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

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