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)=7+2x-5x²-10x⁴

Rewrite 4-5x-x²+6x³ in descending powers of x.

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\(\left\)(x\(\right\))=x^2\(\left\)(x-1\(\right\))^3\(\left\)(2x+6\(\right\))$

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. $f\left(x\right)=4x-x^3$

Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=5x^2+6x^3$

Determine which functions are polynomial functions. For those that are, identify the degree. $h\left(x\right)=7x^3+2x^2+\frac{1}{x}$

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = x² - 4x + 3

For each polynomial function, identify its graph from choices A–F.

$\text{ƒ}\left(x\right)=-(x-2{)}^2(x-5)$

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁴-x³+3x²-8x+8; no real zero greater than 2

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)(x-5)

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=x⁴-7x³+13x²+6x-28; [-1, 0]

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 corner 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.

b. Determine when the volume of the box will be greater than 40 in.³.

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x³+x²−2x+1; between -3 and -2

In Exercises 25–26, graph each polynomial function. $f\left(x\right)=-x^3(x+4{)}^2(x-1)$

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)²(x-5)²