Zeros & Multiplicity
What is the definition of a zero in a polynomial?
A value of x that makes the function equal to 0.
What determines the end behavior of a polynomial?
The degree and leading coefficient.
What is the degree of
f(x)=(x-2)^2(x+3)^3
Degree = 5
A graph bounces off the x-axis at x=−2 and crosses at x=3. Write one possible equation.
f(x)=(x+2)^2 (x−3).
Explain how to tell if a graph “bounces” or “crosses” at a zero.
Even multiplicity → bounces; odd → crosses.
If f(x)= (x - 2)3(x + 4)2, what are the zeros and their multiplicities?
Zeros: x=2 (mult. 3), x=−4 (mult. 2).
Describe the end behavior of
f(x)=-x^3 +4x^2 -x.
-/odd
As
x -> infty, f(x) ->-infty
x-> -infty, f(x)-> infty
If the degree is odd and leading coefficient is negative, what does the graph look like?
right-down, left-up
Which has the steeper end growth:
x^3 or x^5 ?
x^5
Sketch the rough shape of
f(x)=(x+1)^2 (x−3).
Bounces at −1, crosses at 3, right-up and left down.
What happens to the graph at a zero with even multiplicity?
It bounces off the x-axis.
True or False
A polynomial with an even degree always has opposite end behaviors.
False. Even degree = same end direction.
Find the leading term of
f(x)=-3(x-1)^2(x+4)
Leading term:
-3x^3
The graph falls left and rises right. What’s the degree and sign of the leading coefficient?
Odd degree, positive leading coefficient.
What are the x-intercepts of
f(x)=(x−4)(x+5)^2 ?
x=4,−5.
Find all zeros and multiplicities for
f(x)=x^2(x−5)^4.
x=0
(mult. 2),
x=5
(mult. 4).
If degree is 5 and leading coefficient is positive, describe the end behavior.
x->infty,f(x)->infty
x->-infty,f(x)->-infty
For
f(x)=−2(x+3)^2(x−4)^3
identify the degree and describe its end behavior.
Degree = 5 (odd), negative leading coefficient → left-up, right-down.
Describe a graph with 3 real zeros.
A cubic polynomial.
Write a polynomial in factored form with zeros at −1,2,4.
f(x)=(x+1)(x−2)(x−4).
Create a polynomial with zeros at −3,0, and 5 where −3 has the multiplicity of 2.
f(x)=x(x+3)^2(x−5).
Explain how to determine end behavior without graphing.
Use the leading term’s sign and degree.
Determine degree and orientation of
f(x)=2(x-1)(x+2)^3(x-5)^2
Degree = 6, orientation = right-up, left up.
Describe the shape of
f(x)=−x^4 +3x^2 .
Opens downward on both ends, symmetric about y-axis.
For
f(x)=(x−1)^3(x+2)^2
sketch and label intercepts
x=1 (cross), x=−2(bounce).