HONORS Algebra Two, Polynomial Unit Day 2 Name:________________________ Date: _____________
APPEARANCE OF FUNCTIONS AND THEIR ROOTS
The following statements are equivalent:
 k is a zero of the polynomial function, f

(x – k) is a factor of f(x)
 k is a solution of the polynomial equation f(x) = 0
 k is an x-intercept for the graph y = f(x)
Example 1: Sketch: y
 x
2 
4 x

5 y
8
6
4
2
(a) What are the zeros? __________________
(b) What are the factors? ___________________
(c) What is the factored form of y
 x
2 
4 x

5 ? __________________
–8 –6 –4 –2 2 4 6 8 x
–2
–4
–6
–8
(d) How many real solutions does the polynomial have? ____________
(e) Describe the end behaviors of the function: ______________________
Example 2: Sketch: y x
3
4 x
2 y
8
6
4
2
(a) What are the zeros? __________________
(b) What are the factors? ___________________
(c) What is the factored form of y x
3
4 x
2
? __________________
–8 –6 –4 –2
2 4 6 8 x
–2
–4
–6
–8
(d) How many real solutions does the polynomial have? ____________
(e) Describe the end behaviors of the function: ______________________
–8 –6 –4 –2
8
6
4
2
–2
–4
–6 y
2 4 6 8 x
Example 3: Sketch: y
  
4 x
3 
4 x
2
(a) What are the zeros? __________________
(b) What are the factors? ___________________
(c) What is the factored form of y
  
4 x 3 
4 x 2
? ________________
(d) How many real solutions does the polynomial have? ____________
(e) Describe the end behaviors of the function: ______________________
–8

Sometimes, a polynomial equation has a factor that appears more than once. This creates a root of _______________.

A simple zero (with a multiplicity of 1) crosses the x-axis in a straight line.


A zero with an even multiplicity touches the x-axis but does not cross it.
A zero with an odd multiplicity crosses the x-axis and bends at the point where it crosses.
Example 4: Below is the graph of f x x
4 x
3
3 x
2 
5 x

2 .
(a) What is the degree of the function? (What is the total number of solutions of the function?) ________
(b) How many real roots does the function have? __________
(c) How many imaginary roots does the function have? ___________ y
8
6
(d) The graph bends and crosses
The x-axis at ( 1, 0)
4

. The zero
2 at

1
–6 –4 of: _______ .
–2
–2
2 4
–4
–6
–8
(f) What is the factored form of
6 x
(e) The graph touches the x-axis at
(2, 0) . The zero at 2 has a x
4 x
3
3 x
2 
5 x

2 ? __________________________________
Example 5: Below is the graph of f x
 x
5  x
4 
5 x
3  x
2 
8 x

4 .
(a) What is the degree of the function? (What is the total number of solutions of the function?) ________
(b) How many real roots does the function have? __________
(c) How many imaginary roots does the function have? ___________ y
8
6 the x-axis at
4
(d) The graph “bounces off”
2
. The zero at

2
(e) The graph bends and crosses the x-axis at (1, 0) . The zero at 1 has a
–6 –4 –2
–2
2 4 6 x has a multiplicity of: _______ . of: _______ . We also call this
–4
–8
(f) What is the factored form of f x
 x
5  x
4 
5 x
3  x
2 
8 x

4 ? __________________________________

Example 6: If a function is a 3 rd degree polynomial and has a zero at 1 with a multiplicity of 2, and a zero at 3 then what is the factored form of the equation of the function?
Example 7: If a function is a 3 rd degree polynomial and has a zero at 0 with a multiplicity of 3, then what is the equation of the function?
Example 8: Find the multiplicity of the zero(s) at x = 0 and x = 3 in the graph to the right.
–2
5 y
2 4
Example 9: Write the equation of the simplest polynomial function with zeros

4, 0 and 1.
–5
Equation of the polynomial: _____________________________ x
(b) What is the degree of this polynomial? __________
(c) How many real roots does this polynomial have? ___________
(d) How many imaginary roots does this polynomial have? ___________
Example 10: Write the equation of the simplest 4 th degree polynomial function with a double roots at x = 0 and x = 1.
Equation of the polynomial: _____________________________
(b) What is the degree of this polynomial? __________
(c) How many real roots does this polynomial have? ___________
(d) How many imaginary roots does this polynomial have? ___________
Example 11: Write the equation of the simplest polynomial function with zeros

4 and
1
2
.

Algebra Two A.15 Name:________________________ Date: _____________
HOMEWORK - APPEARANCE OF FUNCTIONS AND THEIR ROOTS
12. Sketch: y
  
6 x y
8
6
4
2
(a) What are the zeros? __________________
(b) What are the factors? ___________________
(c) What is the factored form of y
  
6 x ? __________________
–8 –6 –4 –2
–2
–4
–6
–8
2 4 6 8 x
(d) How many real solutions does the polynomial have? ____________
(e) Describe the end behaviors of the function: ______________________
13. Sketch: y
 x
3 
2 x
2 
4 x

8
8
6
4
2 y x
(a) What are the zeros? __________________
(b) What are the factors? ___________________
(c) What is the factored form of y
 x
3 
2 x
2 
4 x

8 ? __________________
–8 –6 –4 –2 2 4 6 8
–2
–4
–6
–8
(d) How many real solutions does the polynomial have? ____________
(e) Describe the end behaviors of the function: ______________________
14. Sketch: f x
 x
4 
9 x
3 
27 x
2 
27 x
–8 –6 –4 –2
–2
–4
–6
–8
8
6
4
2 y
2 4 6 8
(a) What is the degree of the polynomial? _________
(b) What is the number of imaginary solutions? ________
(c) The zero at 0 has a multiplicity of ___________. x
(d) The zero at 3 has a multiplicity of ___________.
(e) What are the roots? _______________________
(f) What is the factored form of the equation? ______________________________
15. Sketch: f x
 x
4 
4 x
2
–8 –6 –4 –2
–2
–4
–6
–8
8
6
4
2 y
2 4 6 8 x
(a) What is the degree of the polynomial? _________
(b) What is the number of imaginary solutions? ________
(c) The zero at

2 has a multiplicity of ___________.
(d) The zero at 0 has a multiplicity of ___________.
(e) The zero at 2 has a multiplicity of ___________.
(f) What are the roots? _______________________
(g) What is the factored form of the equation? ______________________________

16. Write the equation of the simplest 4 th degree polynomial function with roots at 1 and 2 and a double root at x = 0.
Equation of the polynomial: _____________________________
(b) What is the degree of this polynomial? __________
(c) How many real roots does this polynomial have? ___________
(d) How many imaginary roots does this polynomial have? ___________
17. Write the equation in simplest form for a 3 rd degree polynomial function with a single real root at x

2 and where one of the other roots is 2 i .
Equation of the polynomial: _____________________________
(b) What is the degree of this polynomial? __________
(c) How many real roots does this polynomial have? ___________
(d) How many imaginary roots does this polynomial have? ___________
18. If a function has a zero at 2 with a multiplicity of 3, then what is the equation (factored form) of the 3 rd degree polynomial?
19. If a function has a zero at 0 with a multiplicity of 2 and a zero at -1, then what is the3 rd degree equation (factored form) of the function?
20. What is the multiplicity of the zero(s) at x = 0 in the graphs below: y y y (b) y x
5
–2 2 4 x x
–5 x
–10