1. M3D13 Have out: Bellwork: pencil, red pen, highlighter, notebook
Complete #1a on the handout.
1) Determine all the zeros and sketch the graph of y = P(x). Label all intercepts. y a)
Degree: ___
Zeros: ____________ x
Remember: Only graph the real zeros, and make sure the graph fits the criteria that is known for sure.

2. Any other possibilities out there?4
Degree: ___
Zeros: ____________ y x
(–2,0)
(2,0)
What are the criteria we know?
(1) x–intercepts
(0, –100)
(2) y–intercept
(3) end behavior
Any other possibilities out there?
Draw a 4th degree polynomial that fits these criteria.

3. 3 b) Degree: ___ Zeros: ____________ End behavior: y (0,0) x (7,0)
(–7,0)
(7,0)
End behavior:

4. Any other possibilities out there?c) 4
Degree: ___
Zeros: ____________ y x
(–2,0)
(2,0)
What are the criteria we know?
(1) x–intercepts
(2) y–intercept
(0, –36)
(3) end behavior
Any other possibilities out there?
Draw a 4th degree polynomial that fits these criteria.

5. d) 3 Degree: ___ Zeros: ____________ End behavior:y
(0, 8) x
(–4,0)
End behavior:
Be sure to put an inflection point somewhere.

6. e) 4 Degree: ___ Zeros: ____________ End behavior: y (0, 25) (–1,0)
(1,0) x
(–5,0)
(5,0)
End behavior:

7. f) 5 Degree: ___ Zeros: ____________ End behavior: y (0,0) x (4,0)
(–3,0)
(4,0)
End behavior:

8. Remember: When there is a complex zero there is always a _____________.
conjugate pair
For instance:
–5i is paired up with +5i.
(9 + 4i) is paired up with (9 – 4i).

9. Part 2: Find a standard (expanded) form equation of a polynomial with the given zeros.
a) cubic polynomial with zeros x = 2 and i.
There must be a conjugate pair to i : –i
So there are 3 zeros:
x = 2, ±i
x = 2
Any time you see a “±”
you have to square both sides.
x – 2 = 0
Expand using F.O.I.L. or a generic rectangle.
Note: We are finding “a polynomial”, so you do not need to find a specific leading coefficient, “a”.

10. b) quartic polynomial with zeros x = 3 and
degree: 4
All the zeros are:
Any time you see a “±” you have to square both sides.
Expand using F.O.I.L. or a generic rectangle.
Check: This is degree is 4!

11. c) quartic polynomial with zeros x = 4i and
degree: 4
2 zeros here.
There has to be two zeros here.
Expand using F.O.I.L. or a generic rectangle.
Check: This is degree is 4.

12. d) 5th degree polynomial with zeros x = 1, i, 2i.
one real zero
2 zeros
2 zeros
Expand using generic rectangles
Yep, it’s degree 5!
Here is a link to check something like this:
poly expand link

13. Complete the worksheet.

14. Determine the polynomial function P(x)
Determine the polynomial function P(x). Write your answers in factored form.
–1, –1, 3 a)
zeros: __________ y
P(x) = a
(x + 1)2
(x – 3)
4 = a(–2 + 1)2(–2 – 3)
4 = a(–1)2(–5)
(–2, 4)
4 = a(1)(–5)
4 = –5a x

15. Determine the polynomial function P(x)
Determine the polynomial function P(x). Write your answers in factored form.
–4, –1, 3, 3 b) y
zeros: _____________
(1, 5)
P(x) = a
(x +4)
(x +1)
(x – 3)2
5 = a(1 + 4)(1 + 1)(1 – 3)2 x
5 = a(5)(2)(–2)2
5 = a(5)(2)(4)
5 = 40a