Chapter 4: Polynomial and Rational Functions

 

1.      Sketch the graph of f(x) = -x 3 - 3.

 

Intermediate Value Theorem

If f is a polynomial function and f(a) =l= f(b) for a<b, then f takes on every value between f(a) and f(b) in the interval [a, b].          *=l=  - not equal to.

 

2.      Use the intermediate value theorem to show that f(x) = 2x 4 + 3x - 2 has a zero between ½ and ¾.

 

3.      Use the intermediate value theorem to show that f(x) = 2x 3 + 5x 2 - 3 has a zero between -3 and -2.

 

Remainder Theorem

If a polynomial f(x) is divided by x - c, then the remainder is f(c).

 

Factor Theorem

A polynomial f(x) has a factor x - c if and only if f(c) = 0.

 

4.      Use the remainder theorem to find f(c).

f(x) = x 4 - 6x 2 + 4x - 8;      c = - 3

 

5.      Use the factor theorem to show the x - c is a factor of f(x).

f(x) = x 3 + x 2 - 11x + 10;       c = 2

 

 

Synthetic Division

               c            a     b     c     ....     e

                            ab   xc    yd    ....

                      a     x     y     z              w

 

6.      Use synthetic division to find the quotient and remainder if the polynomial is divided by by the second.

2x 3 - 3x 2 + 4x - 5;   x - 2

 

7.      Find a polynomial f(x) in factored form that has degree 3; zeros 2, -1, and 3; and satisfies f(1) = 5.

 

8.      Sketch the graph of f(x) = 4/x.

 

 

9.      Sketch the graph of f(x) = 3/(x - 4).

 

10. Sketch the graph of f(x) = (x - 4)/(x 2 - 6x + 8).