8
Polynomials Worksheet
Concepts:
Quadratic Functions
– The Definition of a Quadratic Function – Graphs of Quadratic Functions - Parabolas – Vertex
– Absolute Maximum or Absolute Minimum – Transforming the Graph of f (x) = x2
to Obtain the Graph of Any Quadratic Function.
Polynomial Functions
– The Definition of a Polynomial – Identifying Polynomials
– Leading Coefficient, Leading Term, and the Degree of a Polynomial – Constant Term of a Polynomial
– Polynomial Division - The Division Algorithm – Roots and Zeros of a Polynomial
– The Remainder Theorem – The Factor Theorem
– The Rational Root Theorem
– The Number of Roots of a Polynomial – Finding Rational Roots of a Polynomial – Finding Real Roots of a Polynomial – Graphs of Polynomials
* Leading Term vs. Shape of the Graph
* Continuous Graphs
* Smooth Graphs
1. Let f (x) = x2
and g(x) = 4x2
− 5x − 6.
Describe the transformations that could be applied to the graph of f to obtain
the graph of g.
Sketch the graph of g.
What is the vertex of the graph of g?
Does the graph of g have an absolute minimum or an absolute maximum? What
is it? 2. Let f (x) = x2
and g(x) = −2x2
+ 3x + 4.
Describe the transformations that could be applied to the graph of f to obtain
the graph of g.
Sketch the graph of g.
What is the vertex of the graph of g?
Does the graph of g have an absolute minimum or an absolute maximum? What
is it?
3. (Exercise 55 from Section 4.1 of your textbook) A field bounded on one side by a river is to be fenced on three sides so as to form a rectangular enclosure. If 200 feet of fencing is to be used, what dimensions will yield an enclosure of the largest possible area?
4. (Exercise 56 from Section 4.1 of your textbook) A rectangular box (with top) has a square base. The sum of the lengths of its 12 edges is 8 feet. What dimensions should the box have so that its surface area is as large as possible.
5. (Exercise 61 from Section 4.1 of your textbook) A potter can sell 120 bowls per week at$5 per bowl. For each 50 cent decrease in price, 20 more bowls are sold. What price
6. (Exercise 59 from Section 4.1 of your textbook) At Middleton Place, a plantation near Charleston, South Carolina, there is a “joggling board” that was once used for courting. A young girl would sit at one end, her suitor at the other end and her mother in the center. The mother would bounce on the board, thus causing the girl and her suitor to move closer together. A joggling board is 8 feet long and an average mother sitting at its center causes the board to deflect 2 inches, as shown in the figure. The shape of the deflected board is parabolic.
2 inches
8 feet
(a) Find the equation of the parabola, assuming that the joggling board lies on the x-axis with its center at the origin.
(b) How far from center of the board is the deflection 1 inch?
7. Find the quotient and the remainder for each of the following division problems. (a) x 3 − 2x 2 + 7x + 1 2x + 3 (b) x 4 + 3x − 1 x2 + 2 (c) 2x + 3 x3 − 2x2+ 7x + 1
8. Find the remainder of the following division problems. (a) x 99 + 7 x − 1 (b) x 100000 + x99999 + 4 x+ 1 x10
10. Find the zeros of the function f (x) = 6x2
− 19x − 36. Use these zeros to help you factor this function.
11. (Exercise 67 from Section 4.2 of your textbook) Use the Factor Theorem to show that for every real number c, (x − c) is not a factor of x4
+ x2
+ 1.
12. Describe the end behavior of each polynomial. Use correct mathematical symbols. (a) P (x) = 2x5 − 3x 2 + 76 (b) Q(x) = −55x100 + 15x75 − 3 (c) R(x) = (2x + 3)4 (50 − x)100
(HINT What is the leading term?) (d) S(x) = (1 − 2x)11
(x + 5)4
13. The graph of a polynomial P (x) is shown below.
500 1000 1500 2000 2500 3000 −500 −1000 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −6 −7 −8 −9 x y
(a) Is the degree of the polynomial even or odd? (b) Is the leading coefficient positive or negative?
(c) What can you say about the factors of this polynomial?
(d) Can you find a formula for the polynomial if you know that the degree of the polynomial is less than or equal to 4 and that P (1) = 24
14. The graph shown below is NOT the graph of y = g(x) = −2(x + 3)(x − 2)(x − 5). Which of the following are clues that this is NOT the graph of g?
(a) The graph crosses the x-axis at (−3, 0), but it should not cross the x-axis at this point.
(b) The graph crosses the x-axis at (5, 0), but it should not cross the x-axis at this point.
(c) The graph has the wrong x-intercepts.
(d) The graph crosses the x-axis at (2, 0), but it should not cross the x-axis at this point.
(e) The graph displays the wrong end behavior.
(f) The graph has too many local extreme points to be the graph of a polynomial of degree 3.
15. The graph shown below is NOT the graph of y = h(x) = 5(x + 1)4
. Which of the following are clues that this is NOT the graph of h?
(a) The graph crosses the x-axis at (−1, 0), but it should not cross the x-axis at this point.
(b) The graph displays the wrong end behavior. (c) The graph has the wrong x-intercepts.
(d) The graph does not have the right number of local extreme points to be the graph of a polynomial of degree 4.
