4. Polynomial Functions

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x³−8x²+x+2; between 2 and 3

Determine which functions are polynomial functions. For those that are, identify the degree. $g\left(x\right)=7x^5-\pi x^3+\frac{1}{5}x$

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

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

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\(\right\))=2+x$

The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. $4x^5$

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/3)(x+3)⁴-3

Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x³+3x²−x−3

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=4x⁷-x⁵+x³-1

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=3(x+5)(x+2)²

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. $f\left(x\right)=x^3-x^2-9x+9$

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x⁴+6x³−18x²; between 2 and 3

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=-5x^4+7x^2-x+9$

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/2)(x-2)²+4