Chapter 5.4, Problem 47HP

To determine

To explain: the leading coefficient and the degree are the only factor at end behavior of polynomial function.

Answer to Problem 47HP

The only factors which matter in determining the end behavior of a polynomial function are the leading coefficient and the fact that the degree is odd or even.

Explanation of Solution

Given:

The leading coefficient and the degree are the only factor.

Concept used:

The function can be expressed in the form of:

  $\text{f}\left(\text{x}\right){\text{=a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n-1}}{\text{x}}^{\text{n-1}}\text{+}\mathrm{......}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}$ .

If $\text{x}\to \infty$ :

Then $\frac{\text{b}}{\text{x}}\text{=0}$ where $\text{b}\ne \infty$ .

Calculation:

The function can be expressed in the form of:

  $\text{f}\left(\text{x}\right){\text{=a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n-1}}{\text{x}}^{\text{n-1}}\text{+}\mathrm{......}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}$ .

  $\text{f}\left(\text{x}\right){\text{=x}}^{\text{n}}\left({\text{a}}_{\text{n}}\text{++}\frac{{\text{a}}_{\text{n-1}}}{\text{x}}\text{+}\frac{{\text{a}}_{\text{n-2}}}{{\text{x}}^{\text{2}}}\mathrm{......}\text{+}\frac{{\text{a}}_{\text{1}}}{{\text{x}}^{\text{n-1}}}\text{+}\frac{{\text{a}}_{\text{0}}}{{\text{x}}^{\text{n}}}\right)$ .

When $\text{x}\to \infty$ :

  $\text{f}\left(\text{x}\right){\text{=x}}^{\text{n}}{\text{a}}_{\text{n}}$ .

There are four Cases:

1. If $\text{n=2k,}{\text{a}}_{\text{n}}\text{<0}$ then $\Rightarrow f\left(x\right)=-\infty$ .
2. If $\text{n=2k,}{\text{a}}_{\text{n}}\text{>0}$ then $\Rightarrow \text{f}\left(\text{x}\right)=\infty$ .
3. If ${\text{n=2k+1, a}}_{\text{n}}\text{<0}$ then $\Rightarrow \text{f}\left(\text{x}\right)=\infty$ .
4. ${\text{n=2k+1, a}}_{\text{n}}\text{>0}$ then $\Rightarrow f\left(x\right)=-\infty$ .

The cases $\text{x}\to \infty$ can be treated as same way.

Hence, the only factors which matter in determining the end behavior of a polynomial function are the leading coefficient and the fact that the degree is odd or even.