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

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=2x⁵-x⁴+2x³-2x²+4x-4; no real zero greater than 1

Graph each polynomial function. ƒ(x)=(x-2)²(x+3)

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=3x⁴+2x³-4x²+x-1; no real zero greater than 1

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

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(4x+3)(x+2)²

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x⁴+x³-6x²-7x-2

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=3x⁴-7x³-6x²+12x+8

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

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-2x(x-3)(x+2)

Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f\(\left\)(x\(\right\))=x^3+1$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x³−4x²+2; between 0 and 1

Determine the end behavior of the given polynomial function. $f\(\left\)(x\(\right\))=x^2+4x+x+7x^3$

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x4+3x³-3x²-11x-6

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 - 2)² - 5