4. Polynomial Functions

Graph each polynomial function. ƒ(x)=-2x⁴+7x³-4x²-4x

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x³-5x²-x+6

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

$\text{ƒ}\left(x\right)=-3x^4-5x^3+2x^2$

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=10x⁶-x⁵+2x-2

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x³+5x²-x-5

Determine the maximum number of turning points for the given polynomial function. $f\(\left\)(x\(\right\))=6x^4+2x$

Graph the following on the same coordinate system.

(a) y = x²

(b) y = 3x²

(c) y = 1/3x²

(d) How does the coefficient of x² affect the shape of the graph?

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

$\text{ƒ}\left(x\right)=x^5+2x^3-2x^2+5x+5$; no real zero less than -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)=2(x−5)(x+4)²

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

The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A certain right triangle has area 84 in.². One leg of the triangle measures 1 in. less than the hypotenuse. Let x represent the length of the hypotenuse. Express the length of the leg mentioned above in terms of x. Give the domain of x.

Graph: f(x) = -2(x − 1)² (x + 3).

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = (x + 3)²

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x³(x²-4)(x-1)

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