Chapter 4.1, Problem 37E

To determine

To write:a polynomial equation of least degree for $-1,1,4,-4,5$ and find the degree as even or odd. Also find the times graph of the related function cross the x-axis.

Answer to Problem 37E

  $x^5-5x^4-17x^3+85x^2+16x-80=0$

The degree of this equation is 5, odd degree

Since all the 5 roots are real, the graph will cross the x-axis 5 times

Explanation of Solution

Given information: $-1,1,4,-4,5$

Calculation:

If $x=-1,$ then

  $x+1\text{ is }$ a factor of the polynomial.

If $x=1,$ then $x-1$ is a factor of the polynomial.

Likewise, if $x=4,x=-4\text{ and }x=5$ then $x-4,x+4\text{ and }x-5$ are factors of the polynomial.

Therefore, the linear factors for the polynomial are

  $\begin{array}{l}\left(x+1\right)\\ \left(x-1\right)\\ \left(x+4\right)\\ \left(x-4\right)\\ \left(x-5\right)\end{array}$

Now find the product of these factors.

  $\left(x+1\right)\left(x-1\right)\left(x+4\right)\left(x-4\right)\left(x-5\right)=0$

Multiply the factors

  $\begin{array}{l}\left(x+1\right)\left(x-1\right)\left(x+4\right)\left(x-4\right)\left(x-5\right)=0\\ \left(x^2-1\right)\left(x^2-16\right)\left(x-5\right)\\ \left(x^4-16x^2-x^2+16\right)\left(x-5\right)=0\\ x^5-5x^4-17x^3+85x^2+16x-80=0\end{array}$

A polynomial equation with roots $-1,1,4,-4,5$ is

  $x^5-5x^4-17x^3+85x^2+16x-80=0$

The degree of this equation is 5. Thus the equation has an odddegree since 5 is an odd number.

Since all the 5 roots are real, the graph willcross the x-axis 5 times