1. Polynomials, Linear Factors, and Zeros
What you’ll learn
To analyze the factored form of a polynomial.
To write a polynomial function from its zeros.
Describe the relationship among solutions, zeros,
x- intercept, and factors.
Vocabulary
Factor theorem, multiple zero,
multiplicity, relative maximum,
relative minimum

2. Now we know that P(x) is a polynomial function, the
solutions of the related polynomial equation P (x)=0
are the zeros of the function. Finding the zeros of the
polynomial will help you factor the polynomial, graph
the function, and solve the related polynomial equation.
Problem 1: Writing a Polynomial in Factored Form
What is the factored form of
GCF: x
Factor
Your turn
What is the factored form of
Answer:

3. Roots, Zeros, and x-intercepts
Take a note
Roots, Zeros, and x-intercepts
The following are equivalent statements about a real
number b and a polynomial
is a linear factor of the polynomial P(x).
b is a zero of the polynomial function y=P(x).
b is a root(or a solution) of the polynomial equation P(x)=0
b is an x-intercept of the graph of y=P(x)

4. Problem 2: Finding zeros of a polynomial function
What are the zeros of ? Graph
the function.
Step 1: Use the zero-product property to find zeros.
Step 2: Find points for x-values between the zeros. Evaluate

5. Step 3: Determine the end behavior.
The function is a cubic function. The coefficient of is 1. So the end behavior is down and up.
Step 4: Use the zeros, the additional
points and end behavior to sketch
the graph.
end behavior is down and up.

6. Your turn
What are the zeros of ?Graph the function
Answer:
(-3,32)
(-5,0)
(0,0)
(3,0)

7. The expression (x-a) is a factor of a polynomial if and only
Factor Theorem
The expression (x-a) is a factor of a polynomial if and only
if the value a is a zero of the related polynomial function
A theorem is a statement that
has been proven on the basis
of previously established
statements, such as other
theorems, and previously
accepted statements.

8. Problem 3: Writing a polynomial function from its zeros
A. What is the cubic polynomial function in standard form
with zeros -2,2, and 3?
Write a linear factor for each zero
Multiply (x-2) and (x-3)
Distributive Property
Simplify
The cubic polynomial has zeros -2,2, and 3.
B. What is a quartic function in standard form with zeros
-2,-2,2 and 3
The quartic polynomial has zeros -2,-2,2, and 3.

9. C. Graph both functions. How do the graphs differ? How are
they similar?
Both have x-int. at -2,2 and 3.
The cubic has down and up end
behavior. The quartic has up
and up behavior. The cubic
function has two turning points
and it cross the x-axis at -2.
The quartic function touches the
x-axis at -2 but doesn’t cross it.
The quartic function has
three turning points.
Note: you can write the polynomial function in the factored form
as and in the g(x)) the repeated linear factor x+2 makes -2 a multiple zero. Since the linear factor x+2 appear twice, you can say that -2 is a zero multiplicity 2. in general, a is a zero of multiplicity n means x+a appears n times as a factor.

10. Your turn
a)What is a quadratic polynomial function with zeros 3 and -3?
Answer:
b) What is a cubic polynomial function with zeros 3,3, and -3?
c) Graph both functions. How do the graphs differ? How are
they similar?
Both graphs have x-int.
of 3 and -3.The quadratic
has up and up and one
turning point, the cubic
has down and up end
behavior and two
turning points

11. Problem 4: Finding multiplicity of a zero.
Note: if a is a zero of multiplicity n in the polynomial function then the behavior of the graph at the x-intercept a will be close to linear if n=1, close to quadratic if n=2, close to cubic if n=3. and so on. If a is a zero of even multiplicity then the graph of its function touches the x-axis at a and turns around. If a is a zero of odd multiplicity then the graph of its function crosses the x-axis at a
Problem 4: Finding multiplicity of a zero.
What are the zeros of ? What are their
multiplicities? How does the graph behave at these zeros?
Factor out the GCF then
factor

12. Since the number 0 is a zero of multiplicity 2.
The numbers -2 and 4 are zeros of multiplicity 1
The graph looks close
to linear at the
x-intercepts -2 and 4.
It resembles a parabola
at x-intercept 0
If the graph of a polynomial function has several turning points, the function can have a relative maximum and relative minimum. A relative maximum is the value of the function at an up to down turning point. A relative minimum is the value of the function at a down to up turning point.
Relative minimum

13. Your turn
What are the zeros of ?
What are their multiplicities? How the graph behave at
this zeros?
Answer:
0 is a zero of multiplicity 1,the graphs
looks close to linear at x=0; 2 is a zero
of multiplicity 2, the graph looks close
To quadratic at x=2

14. Your turn again
What are the relative maximum and minimum of
Answer:
Using a graphing calculator the maximum is 80 at x=-4
And the relative minimum is -28 at x=2
The relative maximum is the greatest
y-value in the neighborhood of the
x-value. The maximum at a vertex
of a parabola is the greatest y-value
of all the x-values

15. Problem 5: Using a polynomial Function to Maximize Volume
The design of a digital camera maximizes the volume
while keeping the sum of the dimensions at 6 inches. If the
length must be 1.5 times the height, what each should each dimension be?
Step 1: Define a variable
Let be x=the height of the camera
Step 2: Determine length and width
Step 3: Model the volume
Step 4: Graph it. If using Graphing calculator use the MAXIMUM feature
to find that the maximum value is 7.68 for a height 1.6

16. A designer wants to make a rectangular prism box with
Your turn
A designer wants to make a rectangular prism box with
maximum volume, while keeping the sum of its length,
width, and height equal to 12 in. The length must be
3 times the height. What should each dimension be?
Step 1: Define a variable
Let
Step 2: Determine length and width
length= width
Step 3: Model the volume
Answer:
Step 4: Graph it.
Height=2
Length= 6
Width= 4

17. Class work odd Homework even
TB pgs exercises 7-44