4. Polynomial Functions

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=-x³-4x²+2x-1

Identify which graphs are not those of polynomial functions.

Solve each problem. A comprehensive graph of ƒ(x)=x⁴-7x³+18x²-22x+12 is shown in the two screens, along with displays of the two real zeros. Find the two remaining nonreal complex zeros.

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=5x^3+7x^2-x+9$

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁵-3x³+x+2; no real zero greater than 2

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=x⁴-4x³-x+3; 0.5 and 1

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$\text{ƒ}\left(x\right)=3x^4+2x^3-4x^2+x-1$; no real zero less than -2

Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x⁴−2x²+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=(x^2+7)/3$

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\(\left\)(x\(\right\))=3x^2+5x+2$

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x(x+1)(x-1)

The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal-size squares from each cor-ner and folding up the sides. Let x represent the length of a side of each such square in inches. Use the table feature of a graphing calculator to do the following. Round to the nearest hundredth.

a. Find the maximum volume of the box.

Use one of the end behavior diagrams below, to describe the end behavior of the graph of each polynomial function.

$\text{ƒ}\left(x\right)=-4x^3+3x^2-1$

Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).]

$f\left(x\right)=x^6-6x^4+9x^2$