1. Polynomial Graphs: Zeroes and Multiplicity
Unit 5
Standard:MM3A1

2. Zeroes Also known as Where the graph crosses the x-axis x-intercept
Solution
Roots
Where the graph crosses the x-axis

3. f(x)=x2 + 2x
What are the zeroes?
How many zeroes?
Degree?
{-2, 0} 2 2

4. g(x) = -2x2 + x What are the zeroes? How many zeroes? Degree? {0, 1/2}

5. h(x)= x3 - x What are the zeroes? How many zeroes? Degree? {-1, 0, 1}

6. j(x) = -x3 +2x2 +3x What are the zeroes? How many zeroes? Degree?
{-1, 0, 3} 3 3

7. k(x)= x4 -5x2 + 4 What are the zeroes? How many zeroes? Degree?
{-2,-1, 1, 2} 4 4

8. l(x) = -(x4 -5x2 + 4) What are the zeroes? How many zeroes? Degree?
{-2,-1, 1, 2} 4 4

9. m(x)=1/2(x5+x4-7x3 -22x2 + 24x) What are the zeroes? How many zeroes?
Degree?
{-4,-3, 0, 1, 2} 5 5

10. n(x)=-1/2(x5+x4-7x3 -22x2 + 24x) What are the zeroes? How many zeroes?
Degree?
{-4, -3, 0, 1, 2} 5 5

11. What is the relationship between degree and number of zeroes?
They are the same
Graph the following polynomial:
y=(x-1)(x-3)
x-intercepts?
Zeroes?
Number of zeroes?
Degree?
{1, 3}
1, 3 2 2
(x-1)(x-3)=x2–x-3x+3 = x2-4x+3
Zeroes and degree

12. (x-1)(x-3)=x2–x-3x+3 = x2-4x+3
Standard form
Factored form
To find the zeroes, set each factor =0
(x-1)=0
X=1
(x-3)=0
X=3

13. Write the standard form of the equation for the polynomial function with the given zeroes
Example:
Steps:
0, 4 and -2
(x-0)(x-4)(x+2)
f(x) = x(x-4)(x+2)
f(x) = x(x2 +2x-4x-8)
f(x)= x(x2 - 2x - 8)
f(x) = x3 - 2x2 – 8x
Use the zeroes to write the factors: (x - the zero)
Write as a function in factored form.
Multiply (Foil) and simplify

14. Write the standard form of the equation for the polynomial function with the given zeroes
1, 3(multiplicity 2)
f(x) = (x-1)(x-3)(x-3)
f(x) = (x2 -3x – x +3)(x-3)
f(x) = (x2 - 4x + 3)(x–3)
f(x) = x3 - 3x2 - 4x2 + 12x +3x– 9
f(x) = x3 - 7x2+15x-9
Multiplicity means that a zero is used as a factor more than once

15. For each function, determine the zeroes and their multiplicity
y=(x-3)(x+2)2
x-3=0
x=3
x+2=0
x=-2 with multiplicity 2
3, -2(mult. 2)
For each function, determine the zeroes and their multiplicity

16. For each function, determine the zeroes and their multiplicity
2. y=x(x-5)10 (x+4)2
x=0
x-5=0
x=5 with multiplicity 10
x+4=0
x=-4 with multiplicity 2
0, 5 (mult 10), -4(mult. 2)
For each function, determine the zeroes and their multiplicity