4. Polynomial Functions

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between -1 and 0

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between 2 and 3

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 Find the zero in part (b) to three decimal places.

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x³-37x²+50x+60 between 7 and 8

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x³-37x²+50x+60 between 2 and 3

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x⁴−5x³−x²−6x+4

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. $f\left(x\right)=x^3+3x^2-4$

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 4x⁴−x³+5x²−2x−6=0

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=4x³−8x²−3x+9

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x⁵+2x⁴−15x³−10x²+12x+8