4. Polynomial Functions

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

$\text{ƒ}\left(x\right)=x^5-3x^3+x+2$; no real zero less than -3

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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)=2x³-5x²-x+1; [-1, 0]

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)=2x³-5x²-x+1; [1.4, 2]

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)=x³+4x²-8x-8; [-3.8, -3]

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.

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.

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

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.

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

Use (2x³−3x²−11x+6)/(x−3)=2x²+3x−2 to factor 2x³-3x²-11x+6 completely.

Graph: f(x) = -2(x − 1)² (x + 3).

Graph the following on the same coordinate system.

(a) y = (x - 2)²

(b) y = (x + 1)²

(c) y = (x + 3)²

(d) How do these graphs differ from the graph of y = x²?