Polynomials and Rational Functions

Find the vertex of the parabola :

1. x² + 3x + 3

Graph by hand by finding the vertex, axis of symmetry, x and y intercepts, leading
coefficient , and extra points. Give the domain and range of the function:

2. x² – 7x + 12

Find the zeros of the function , state the multiplicity, and state if the line crosses or touchs
the x-axis:

3. 2x³ – 4x² + 2x

Prove the function has a zero within the interval by using the Intermediate Value
Theorem:

4. f(x) = x⁴ – x³ + 3x² – 5; (0, 2)

Divide using long division:

5. (6x² + 3x⁴ – 3x + 2) ÷ (x + 3)

Divide using synthetic division :

6. (7x – 5x⁵ + 4x² –8) ÷ (x + 3)

List all possible zeros and find all zeros of the function.

7. x³ – x² – x – 2 = 0

Find all zeros of the function by using the Rational Zero Theorem, Descartes' Rule of
Signs and quadratic formula .

8. 4y⁵ + 12y⁴ – 41y³ – 99y² + 10y + 24 = 0

Graph the function by finding horizontal, vertical, and/or slant asymptotes, symmetry, x
and y intercepts , and extra points if possible.

Solve the inequality and write answer in interval notation.

Solve :

14. The cost of a product is a function of the number of products produced as given in the
equation where x is the number of products produced and C(x) is in dollars.

a. Find and interpret C(20), C(40), and C(60).
b. Where is the vertical asymptote?
c. Graph the function .