Unit 3 – Polynomial Functions
What is a polynomial?
Graphs of polynomial functions are _______________________and have smooth, _____________
turns.
Degree of a polynomial –
Left and Right Hand Behavior
If “n” is even…
If “n” is odd…
F(x) = xn
As n approaches infinity…
Leading Coefficient Test…
Describe the left and right hand behavior and max
# of turns for these functions
1. f(x) = 7x3 + 2
2. f(x) = 2x4 – 3x + 5
3. f(x) = -5x5 + 9
Number of turns
Practice:
4. f(x) = -2x4 + 16x – 12
Zeros of Polynomial Functions
Three other terms for zeros:
For a polynomial function, f, of degree n …
1.)
1.)
2.)
2.)
3.)
Let f(x) be a polynomial function and a € R
1.
2.
3.
4.
Find the zeros of the given function
1. f(x) = 2x3 + 8x2 – 42x
2. f(x) = x4 + x3 -6x2
Now we have a better idea about the graph of these functions:
f(x) = 2x3 + 8x2 – 42x
f(x) = x4 + x3 -6x2
Multiplicity
Write a polynomial with the following solutions:
3.) 4, -5, 0
Assignment:
4.) 1, 1, -3
End behavior and x-intercepts
Sketch a graph with the shown end behavior and x-intercepts
1.
2.
3.
4.
5.
6.
7.
8.
9.
Find the zeros of the following functions:
f ( x)  x 2  8 x  16
1 2 5
3
x  x
2
2
2
1. f ( x)  49  x 2
2.
4. h( x)  x 2  x  2
5. g ( w)  w 2  10w  25
6. f ( x)  x 2  25
7. f ( x)  5 x 2  10 x  5
8. f ( x)  x 4  x 3  20 x 2
9. f ( x)  x 5  x 3  6 x
3.
f ( x) 
Sketch the graph of the following using end behavior and x intercepts
10. f ( x)   x 2  10 x  16
11. f ( x)  x 3  3x 2
12. f ( x) 
1 3
x ( x  4) 2
3
Long division with polynomials:
Warm up: perform the long division by hand:
15,627 ÷ 50
Long division with polynomials is done the same way:
1. (x – 10x – 2x + 4)/(x + 3)
4
2
2. (6x3 – 19x2 + 16x - 4) divided by (x - 2)
3. (x3 – 1) / (x-1)
Key step
Synthetic Division
Used only with __________________________________________________________
4. (x4 – 10x2 – 2x + 4) divided by (x + 3)
5. (3x3 – 16x2 – 72) / (x - 6)
Assignment:
Further steps in synthetic division
Be careful with the order of the polynomial
Given one factor of a polynomial
Practice: Use synthetic division to show that x is a factor of the polynomial equation and use the
result to factor the polynomial completely.
1.)
2.)
Assignment:
Terminology Time:
Real zeros of a polynomial
Descartes’ Rule of Signs
Used to determine the number of Real Zeros.
(Note: variation in sign means a change in sign from one term to the next)
Let f(x) = anxn + an-1xn-1+…+ ax + a, be a polynomial f’n with real coefficients and an ≠ 0…
1. The number of positive Real solutions is either equal to the number of variations in the signs
of f(x) or less than that number by an even integer.
2. The number of negative real solutions is either equal to the number of variations in the signs
of f(-x) or less than that number by an even integer.
In summary….
Examples:
Use Descartes’ Rule of Signs to determine the number of + and – real solutions of f(x)
1.) f(x) = -5x3 + x2 – x + 5
Rational Zero Test
2.) g(x) = 2x3 – 3x2 – 3
Examples:
Find the possible rational zeros of f:
Solve the equation:
1. f(x) = x3 + 3x2 – x – 3
f(x) = x3 + 3x2 – x – 3
Find the possible rational zeros, and solve:
2. g(x) = x3 – 6x2 + 11x – 6
Assignment:
Unit 3 Review
1.) Given the polynomial 𝑔(𝑥) = 2𝑥 6 + 4𝑥 4 − 𝑥 3 − 10𝑥 2 + 9, determine the number of possible
positive, negative, and complex zeros it will have. Then explain how these were determined.
2.) Describe the left and right end behavior of (𝑥) = 2𝑥 7 − 3𝑥 3 − 8𝑥 + 2 , and explain how this is
determined.
3.) Find all of the following information for the polynomial𝑓(𝑥), then sketch the graph.
𝒇(𝒙) = 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟏𝟎𝒙 − 𝟐𝟒 and 𝒙 + 𝟐 is a factor of 𝑓(𝑥).
List all zeros:
End Behavior:
4.) Find all of the following information for the polynomial𝑓(𝑥), then sketch the graph.
𝒇(𝒙) = −𝒙𝟒 + 𝟓𝒙𝟑 + 𝟗𝒙𝟐 − 𝟒𝟓𝒙
# pos zeros possible
# neg zeros possible
Possible zeros to try:
Actual zeros of function:
5.) Determine the number of positive real solutions, negative real solutions, and complex solutions
using Descartes Rule of Signs. Then use the rational zero test to list all possible zeros of the
function. Then find all zeros of the function.
𝑓(𝑥) = 2𝑥 4 − 3𝑥 3 + 33𝑥 2 − 48𝑥 + 16
Possible # of Positive Zeros:
Actual Zeros of 𝑓(𝑥) found by solving:
Possible # of Negative Zeros:
Possible # of Complex Zeros:
Possible Rational Zeros to try:
6.) Write a polynomial, in definition form, with the following zeros:
𝑍𝑒𝑟𝑜𝑠: − 2(𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑦 2), 𝑎𝑛𝑑 3𝑖