Name _____________________________________________                           

Honors Math 3 Unit 4: Polynomial Functions

Monday Tuesday Wednesday Thursday Friday

March 5

Long Division

HW: worksheet 4.1

March 6

Synthetic Division; Remainder and Factor

Theorem

HW: worksheet 4.2

March 7

Zeros and Multiplicity

HW: worksheet 4.3

March 8

End Behavior & Function Analysis

HW: worksheet 4.4

March 9

QUIZ!

HW: worksheet

4.5

March 12

Solving Polynomials Completely

HW: worksheet 4.6

March 13

Writing Polynomials from Roots

HW: worksheet 4.7

March 14

Applications and Average Rates of Change

HW: worksheet 4.8

March 15

Review

March 16

Test!!!

Homework 4.1

POLYNOMIAL DIVISION HOMEWORK

Answer the questions, showing all work, on a separate page

1) Find the quotient and remainder, then write the division statement for each polynomial division.

a) 9203913 23 xxxx f) 871510 23 xxxx

b) 237823 xxxx g) 126134 3 nnn

c) 13535 23 xxxx h) 1272425 223 xxxxx

d) 667112 23 aaaa i) 125571110 2234 aaaaaa

e) 220123 xxx j) 525101346 2234 ttttt

2) One factor of 3031154 23 xxx is 2x . Completely factor 3031154 23 xxx .

3) Two factors of 12883912 324 aaaa are 2a and 12 a . Find the other factors.

4) When 1010 23 xmxx is divided by 35 x , the quotient is 22 2 nxx and the remainder is 4. Find the values for m and n.

Homework 4.2 PRACTICE: Synthetic Division, Remainder & Factor Theorem. Solve given a zero

Divide using Synthetic division:

1) (2x3 – 3x2 – 4x + 5) ÷ (x + 1)

2) (8x2 – 26x – 9) ÷ (x - 7)

3) (9x3 + 18x2 – 4x -10) ÷ (x + 2)

4) (x6 – 4x3 - 42) ÷ (x - 1)

5) (2x4 – 5x3 + 2x2 + 5x – 10) ÷ (x – 2)

6) (x3 – 4x2 + 9) ÷ (x – 3)

7) (x4 – 2x3 – 70x + 20) ÷ (x – 5)

8) (4x4 + 5x3 + 2x2 – 1) ÷ (x + 1)

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Use the Remainder & Factor Theorems:

1) What is the remainder when 3x3 + 10x2 + x – 6 is divided by x + 3?

2) Is (x – 2) a factor of 4x2 + 13x + 10? Explain how you know.

3) What is the remainder when 3x3 + 10x2 + x – 6 is divided by x – 1?

4) Is (x – 1) a factor of x3 + 2x2 – 2x – 1? Explain how you know.

5) Is (x + 2) a factor of 4x2 + 13x + 10? Explain how you know.

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Find the zeros using the given information:

1) Find all the zeros of f (x)=x3−4 x2+x+6 given that x + 1 is a factor.

2) Solve for all the solutions of 2 x3−5x2+x+2=0 given that 2 is a solution.

3) Find all the zeros of g(x )=2 x3+3 x2+8 x+12 if −32 is a root.

Homework 4.3

1. Fill in the missing information.

Polynomial Function Name (degree) Name (terms) End Behavior

f ( x )=3 x2−5

y=−x4+6 x−1

g ( x )=6 x

h ( x )=5 x2−2 x3+7 x−3

2. Identify the zeros of each function below. Be sure to state any multiplicity.

3. Use the given information to complete the missing columns.

Table of Values Graph Key Features of the Function

X Y

The y-intercept is (0, 7). The zeros are located at x = 4 and x = 7. There is a relative minimum at (-5.5, 1.5) and at (5.5, -2.5). A relative maximum is located at (-1, 8.5). The polynomial is quartic.

4. Given the graph, state the intervals where the graph is increasing/decreasing and where the graph is positive/negative.

X Y

Homework 4.4 Polynomial Graph MatchingThis worksheet consists of 6 graphs, 6 functions, 6 descriptions of “end behavior”, and 6 descriptions of the roots. Your task is to create 6 sets- each set matches a function with its graph, the description of its roots, and its end behavior that matches that function. The scales on the pictures aren’t all the same, but you can use what you know to connect each of these pieces of information.

1) y = .8x2 + 3x - 8 2) y = 4.5x - 3 3) y = -2x5 + 4x4 + 3x3 - 5x2 + 6x + 20

4) y = (x+2)(x-3)(x+5)(x-8) 5) y = x3 - 8x2 + 10x + 10 6) y = -2x3 + 4x2 + 4x + 15

# of zeros: End Behavior:

4 real roots 3 real roots 1 real root, 2 imaginary roots

(∞, -∞) (∞,∞) (-∞,∞)

1 real root, 4 imaginary roots

1 real root 2 real roots (∞, -∞) (-∞,∞) (∞,∞)

Homework 4.5

Transformations of Polynomial Functions Homework1. Write the equation for the graph of function g(x), obtained by shifting the graph of  f  (x) = x²  three units

left, stretching the graph vertically by a factor of two, reflecting that result over the x-axis, and then translating the graph up four units.

2. Describe the transformations that would produce the graph of the second function from the graph of the first function.

a. f ( x )=x2 becomes f ( x )= (x−3 )2+5

b. f ( x )=x3 becomes f ( x )=−3x3−1

c. f ( x )=x4 becomes f ( x )=12

( x+1 )4−3

d. f ( x )=x2 becomes f ( x )=−2 (3 x−2 )2+5

3. Write the equation for the graph of function g(x), obtained by shifting the graph of  f  (x) = x4  two units right and up four units.

Homework 4.6 – Solving Polynomial

Solve each polynomial.1. x4−5x2−36=0 2. x3−2 x2+3 x−6=0

3. x3+3 x2−14 x−20 4. x3−5 x2−4 x+2=0

Homework 4.7 Polynomial Word Problems Homework

1. At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters. The block’s dimensions are x meters high by 12x – 15 meters long by 12x – 21 meters wide. What is the height of the block?

2. You are designing a chocolate mold shaped like a hollow rectangular prism for a candy manufacturer. The mold must have a thickness of 1 cm in all dimensions. The mold’s outer dimensions should also be in the ratio 1:3:6. What should the outer dimensions of the mold be if it is to hold 112 cubic centimeters of chocolate?

3. A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about 89.58 cubic feet. The tank’s walls will be one half inch thick and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follows:

The width should be 2 feet less than the length The height should be 8 fee more than the length

What should the outer dimensions of the tank be?

4. From 1985 to 2003, the total attendance A (in thousands) at NCAA women’s basketball games and the number T of NCAA women’s basketball teams can be modeled by A=−1.95 x3+70.1 x2−188 x+2150 and

T=14.8 x+725 where x is the number of years since 1985. Compare and contrast the two functions. Find the attendance and number of teams for the year 1998.

5. Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to be 5 inches less than the length. What should the dimensions of the prism be if you want to use all of your clay?

Homework 4.8 Comparing Average Rates of ChangeThe average rate of change of a function that is non-linear will not be constant. It can be positive or negative, whole number or fraction.

Consider the parabola shown to the right. To find the average rate of change between x = 1 and x = 2, you need to locate the ordered pairs that correspond to those points on the graph. The same formula used for slope can be used to find the average rate of change.

(1, 0) and (2, -3)

If you change your x values to x = 4 and x = 6, you will see the average rate of change will not be the same as it was above.

You Try!

Find the average rate of change for the following function over the given interval. What does the average rate of change tell you about the function on the interval?

a)

b)

Unit 4 Polynomial Functions Review

1. Complete the table below

Function Degree End Behavior

Domain and Range

A f ( x )=3 x5−x10

B g ( x )=−x2+5x+3

C h ( x )=3(x+2)(x−4)

D j ( x )=−2 x3−x2+5 x−1

2. Evaluate the polynomial f ( x )=3 x5−x3+6 x2−x+1 for x=−2. Explain what your answer represents.

3. Find the zeros for the function f ( x )=x3+3 x2−x−3

4. Show whether -4 is a zero of g ( x )=x3−x2−14 x+24

5. Use the graph to answer the following questions

a) Relative maximum:

b) Relative minimum:

c) Increasing interval:

d) Decreasing interval:

e) Domain:

f) Range:

g) End Behavior:

h) Zeros:

i) Factored Form:

j) Degree

Find all the zeros

6. f ( x )=2 x3+3 x2−39 x−20 and 4 is a zero

7. f ( x )=x4+3 x2−4 and 1 is a zero

Divide using long division

8. x3−3 x2+8 x−5÷ ( x−1 )

9. 4 x3−12 x2−x+15 ÷ (2x−3 )

10. Sketch a graph f ( x )=−4 ( x−1 )2(x−3)(x+8)

11. Write the function of x4 shifted 3 units down, 4 units left, a

reflection over the x-axis and a horizontal compression by 3.

12. A cement walk of uniform width surrounds a rectangular swimming pool that is 10 m wide and 50 m long. Find the width of the walk if its area is 896 m2.

13. The number of eggs, f(x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by f(x) = 14x3 – 17x2 – 16x + 34. What is the abdominal width when there are 211 eggs?

14. A pyramid can be formed using equal-size balls. For example, 3 balls can be arranged in a triangle, then a fourth ball placed in the middle on top

of them. The function p(n) = 16 n(n+1)(n+2) gives the number of

balls in a pyramid, where n is the number of balls on each side of the bottom layer. (For the pyramid described above, n = 2. For the pyramid in the picture, n = 5.)

a. Evaluate p(2), p(3), and p(4). Sketch a picture of the pyramid that goes with each of these values. Check that your function values agree with your pyramid pictures.

b. If you had 1000 balls available and you wanted to make the largest possible pyramid using them, what would be the size of the bottom triangle, and how many balls would you use to make the pyramid? How many balls would be left over?